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An Integrated Supply-Demand Approach to Solving Optimal Relocations in Station-
based Carsharing Systems
Sisi Jian a,*, David Rey a, Vinayak Dixit a
a Research Centre for Integrated Transportation Innovation (rCITI), School of Civil and
Environmental Engineering, University of New South Wales Australia, Sydney, New South
Wales 2052, Australia
*Corresponding author. Tel.: +61 422 430 935; Fax: +61 (2) 9385 6139; Email address:
s.jian@unsw.edu.au
Abstract
The dominant challenge in one-way carsharing systems is the vehicle stock imbalance. Previous
studies have proposed relocation approaches to handle it using optimization and simulation
models. However, these models do not consider the interdependence between supply and
demand in carsharing systems. In this paper, we develop an integrated optimization model to
link supply and demand together. A discrete choice model that includes vehicle availability as
a parameter directly affecting user’s mode choice is introduced and incorporated within the
optimization formulation. In this framework, carsharing travel demand is influenced by vehicle
supply. The reaction of the demand further changes vehicle availability in the system. The
incorporation of a discrete choice model with the Integer Linear Programming formulation
leads to a nonlinear model. We propose a linearization scheme to reformulate it. We test the
model in realistic case studies representative of an Australian carsharing operator. A sensitivity
analysis on total travel demand, system capacity, one-way trip price, and vehicle availability
coefficient is undertaken to evaluate their impacts on system profit. The results reveal that the
pattern of profit over trip price varies across scenarios with different vehicle availability
coefficients and travel demand. The profit-efficiency of enlarging carsharing network is also
dependent on travel demand. We conclude that the interdependence between demand and
supply should be considered when setting network development plans and pricing strategies in
one-way carsharing systems. If there is a strong interaction between demand and supply, the
supply of carsharing vehicles has a critical impact on system profit.
Keywords: Carsharing; Integer Linear Programming; Supply and Demand; Discrete Choice
Model
Acknowledgements
The authors would like to thank Mr. Bruce Jeffreys and Ms. Rachel Moore from GoGet for
providing data. We would also like to thank the Australian Research Council for their support
under Linkage Grant # LP130100983.
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1 Introduction
The increase of automobile ownership has brought significant issues to the sustainable
development of urban transport, such as traffic congestion, air pollution, and low road resource
utilization. To mitigate these negative externalities, urban carsharing systems represent
effective alternatives (Millard-Ball, 2005, Katzev, 2003). In carsharing systems, the operator
locates a fleet of vehicles to designated vehicle pods. Users are free to pick up vehicles at any
station and return them for a relatively shorter time period compared to traditional car rental,
usually by the hour or by the minute. From the perspective of individual travelers, carsharing
systems offer travelers higher flexibility than public transport, reduce their traveling costs
compared to private vehicles, and provide them comparable travel experience simultaneously.
Moreover, studies have shown that carsharing has a positive impact on reducing vehicle
ownership (Cervero et al., 2007, Prettenthaler and Steininger, 1999). As reported by Ter Schure
et al. (2012), the average vehicle ownership among carsharing member households was 0.47 as
compared to 1.22 among non-carsharing member households in San Francisco. The decrease
in vehicle ownership implies more efficient use of road resources, less congested in urban
transport networks, lower transport costs for urban society, and lower pollutions to the
environment. These attractive features of carsharing systems boost their development around
the world. By 2013, carsharing has spread to over 27 countries around the world with an
estimated 1,788,000 carsharing members sharing over 43,550 vehicles (Shaheen and Cohen,
2013).
Carsharing systems are categorized into different forms based on vehicle return policies.
The most common and traditional form is round-trip carsharing that requires users to return the
vehicles to the same places where they pick them up. This form of carsharing systems is simple
for operators to maintain but not always convenient for travelers. A more flexible system type
is the one-way station-based carsharing which allows users to return their vehicles to any
carsharing stations close to their destinations. The flexibility of such systems is better suited to
users who only need to make one-way trips, such as leisure, shopping, and sporadic trips.
Therefore, one-way carsharing systems have the potential to capture a substantially larger
market share than round-trip carsharing systems. The market attractiveness of one-way
carsharing systems and the fierce competition in carsharing industry have encouraged a number
of companies to provide one-way carsharing services to users, such as Autolib in France
(Autolib.eu, 2016), Zipcar and Car2go in U.S. (Zipcar.com, 2016), and Communauto in Canada
(Communauto.com, 2016). As for the city of Sydney, Australia, over 15,000 travelers have
joined the carsharing schemes since 2003 (Cityofsydney.nsw.gov.au, 2016). None of the
operators provide one-way trip services to the users presently. However, the potential for
attracting a substantially larger market demand of one-way carsharing has encouraged Sydney
carsharing operators to consider one-way trip services to the existing round-trip carsharing
systems.
Along with its potential benefits, the flexibility of one-way carsharing also brings more
complex problems to the operators. One dominant challenge is to ensure the supply of
carsharing vehicles can meet the demand of carsharing users. Since travelers’ origins and
destinations are not necessarily uniformly distributed across urban networks, carsharing vehicle
stocks can become spatially and temporally imbalanced. This may influence vehicle availability.
A possible situation can be users’ pick-up reservations cannot be fulfilled due to a lack of
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vehicles, and users’ parking needs cannot be satisfied due to surplus vehicles. Such poor
accessibility may considerably impair the profitability of one-way carsharing systems. Hence,
it is critical to dynamically relocate vehicles to rectify and anticipate vehicle imbalance.
Relocation operations represent an increase in operational costs, which could potentially be
compensated with the revenues from the increased demand for one-way trips or, in turn, could
reduce benefits. Jorge et al. (2015a) proposed a model to evaluate the profitability of
introducing one-way carsharing services to the round-trip carsharing system in Logan Airport.
They concluded that the supplementary one-way services could be profitable, but the operators
needed to develop efficient relocation strategies. This result highlights the importance of
conducting cost-benefit analysis before adding one-way carsharing services, and the necessity
of developing efficient relocation models.
Under such circumstances, several research efforts have been focused on building effective
relocation strategies for one-way carsharing systems. The proposed strategies are mainly
categorized into operator-based relocation methods (Fan et al., 2008b, Nair and Miller-Hooks,
2011, de Almeida Correia and Antunes, 2012, Jorge et al., 2012, Barth and Todd, 1999, Kek et
al., 2006, Kek et al., 2009, Weikl and Bogenberger, 2015, Nourinejad and Roorda, 2014) and
user-based relocation approaches (Barth et al., 2004, Uesugi et al., 2007, Di Febbraro et al.,
2012). Operator-based relocation requires extra staff to redistribute vehicles between stations,
while user-based relocation encourages carsharing users to relocate vehicles by offering them
price incentives. A detailed literature review with respect to one-way carsharing relocation
methods can be found in Jian et al. (2016), where we found most relocation models did not
consider the impact of demand variation on the availability of carsharing vehicles as well as
vehicle stock distribution. Fan et al. (2008a), Nair and Miller-Hooks (2011), Nair et al. (2013),
Jorge et al. (2012) and Jorge et al. (2015b) considered stochastic travel demand in the relocation
models. They found that demand variation would impair the performance of relocation models.
Jorge and Correia (2013) also highlighted in their literature review paper that it is difficult to
accommodate demand variation when developing relocation optimization models because the
demand and the supply in carsharing systems are interdependent. Users’ travel demand for
carsharing services is influenced by the available supply of carsharing vehicles. Demand, on
the one hand, brings about the revenues to carsharing operators; Relocation, on the other hand,
generates extra costs to them. These two factors contribute in determining the profitability of
carsharing systems. Hence, the interdependence between demand and supply in one-way
carsharing systems makes the profitability analysis more complex.
An attractive solution to this logistical challenge is to undertake an integrated supply and
demand cost-benefit analysis before introducing one-way carsharing services. Jian et al. (2016)
proposed an approach that incorporates an Integer Linear Programming (ILP) model with a
discrete choice model. In their study, users’ travel demand is determined by the discrete choice
model which considers trip cost and travel time to be the main factors influencing users’ demand.
Vehicle availability together with users’ travel demand are the inputs of the ILP model. The ILP
model solves the optimal relocation decisions and determines which trips to accept. The outputs
further change vehicle availability across carsharing vehicle pods. The ILP model and the
discrete choice model work together to account for the interaction between users’ travel demand
and the supply of vehicles in carsharing systems. However, it should be noted that the choice
model cited by the study itself does not consider vehicle availability as a factor that could
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directly influence the elasticity in demand. The integrated approach treats vehicle availability
as a constraint to incorporate the impact of vehicle availability on trip demand. But if we aim
to understand the interaction between demand and supply in carsharing system, the best solution
is to incorporate the ILP model with a choice model that considers users’ demand to be directly
elastic to vehicle availability.
In this paper, we build on the model proposed by Jian et al. (2016) and present a refined
integrated formulation for one-way carsharing optimization. In the new integrated supply-
demand model, vehicle availability is the output of the ILP model and serves as the input of the
discrete choice model simultaneously. In this way, user demand and vehicle supply are linked
by vehicle availability.
However, another difficulty arises in the integrated supply-demand model owing to the
nonlinearities induced by the integration of the discrete choice model, because vehicle
availability is a decision variable in both the ILP model and the discrete choice model. A similar
issue has been found in the automated vehicle assignment problem studied by de Almeida
Correia and van Arem (2016). In their study, travelers’ mode choice between public transport
and automated vehicles was supposed to be determined by a choice model with a Logit or Probit
structure. But in order to avoid the nonlinearity issue, they ignored the probabilistic part of the
choice model and only considered the deterministic part. In another carsharing study conducted
by Jorge et al. (2015b), the authors assumed carsharing demand to be elastic to trip price. Based
on this assumption, they proposed a pricing model to balance vehicle stocks. The price elasticity
of demand caused the model to be nonlinear. They proposed an iterated local search (ILS)
metaheuristic to solve the nonlinear model. These two studies either simplify the choice model
to avoid the nonlinearity, or proposed heuristic to solve the nonlinearity.
Apart from these two approaches, nonlinear formulations can be linearized and converted
into linear models in some cases, at the expense of auxiliary decision variables. The linearized
model can be conveniently solved using robust methods such as LP-based branch and bound
method, and implemented in codes such as CPLEX and XPRESS (Grossmann, 2002). We apply
this technique to the present integrated logistical problem. To the authors’ best knowledge, this
paper is the first to use linearization method to solve a Mixed Integer Nonlinear Programming
(MINLP) model embedding a discrete choice model.
The paper is organized as follows. In the next section we present the extended integrated
supply-demand model derived from the ILP model developed by Jian et al. (2016). Then, the
linearization method of the model is described. The Sydney carsharing network case study is
presented in the fourth section, followed by the results of the sensitivity analysis. The major
findings obtained in this study and the recommendations for further studies are highlighted in
the last section.
2 Integrated Supply-Demand Model
The problem addressed in this paper involves the introduction of one-way trip services to the
existing round-trip carsharing system, while keeping the locations and the number of carsharing
vehicle pods constant. The added one-way services may attract more carsharing travel demand.
The demand is determined by vehicle availability, travel cost and travel time. The demand also
influences vehicle availability and changes the distribution of vehicle stocks in the network.
Hence, the updated carsharing system may incur extra relocation costs to balance the vehicle
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stocks. The objective of this integrated supply-demand model is to determine the optimal
relocation decisions in order to maximize the profit for the updated carsharing system. We make
the following assumptions for this problem:
1. We assume discrete time steps during which decisions are made;
2. All the vehicles are available at the initial time step of each day, and the number of vehicles
in each vehicle pod is equal to the capacity of that pod;
3. Users can only make a trip request when a vehicle is available. Since the supply of vehicles
is limited, not all carsharing trip requests can be accepted by the operator. The operator
will determine the most profitable trips to be accepted;
4. The total travel demand of each origin-destination pair at each time step is known and
constant. However, the operator does not know the demand for carsharing trips in advance.
Each user has three travel mode options: one-way carsharing (OW), round-trip carsharing
(RT), and other transport services (TS). User’s choice is influenced by trip price, trip travel
time, and the number of vehicles available in carsharing vehicle pods. A multinomial logit
(MNL) model is applied to account for trip price, travel time, and the number of vehicles
available. The parameters of the MNL model include trip travel time (in number of time
steps), travel cost, parking time, the number of household cars for each user, and the
number of vehicles available in carsharing vehicle pods;
5. If a vehicle is used for a round trip, the parking space for that vehicle is reserved until this
trip ends; while if a vehicle is used for a one-way trip, its parking space is available;
6. Carsharing users are assumed to be charged by the travel time.
The optimization model and the mathematical notations used to formulate the model are
presented as the following:
Sets
Let be a complete graph, where is a set of nodes and is a set of arcs, i.e.
. Let be the set of time steps in the operation period, and be the set of time
steps excluded the first time step.
Decision variables
: Integer variable that represents the number of vehicles relocated from node to at
time , ;
: Integer variable that represents the number of vehicles available at node i at time t,
;
: Integer variable that represents the number of users booking one-way trips from node
to at time , ;
: Integer variable that represents the number of users booking round trips from node
to at time , .
: Binary variable, if is equal to , ; otherwise ,
.
Parameters
: Price rate per time step for one-way trips;
: Price rate per time step for round trips;
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: Cost of maintaining a vehicle per time step driven;
: Cost of relocating a vehicle per time step driven;
: Capacity of node , ;
: Total travel demand from node to at time , ;
: Travel time from node to (in number of time steps), ;
: Time that a round-trip user spends at destination node , ;
: Probability that a user traveling from node to chooses RT among RT, OW and TS,
, ;
: Probability that a user traveling from node to chooses OW among RT, OW and TS,
, ;
: A parameter changing from 0 to the capacity of the corresponding pod (), ;
: Probability that a user traveling from node to chooses RT among RT, OW and TS
when the number of vehicles available at node is equal to , , , ;
: Probability that a user traveling from node to chooses OW among RT, OW and TS
when the number of vehicles available at node is equal to , , , ;
: Parking time of each trip;
: Number of household cars for each user;
: Coefficient for ;
: Coefficient for travel time;
: Coefficient for travel cost;
: Coefficient for parking time;
: Coefficient for the number of household cars for each user;
: Constant for carsharing;
: Constant for other transport services.
Objective function
(1)
Subject to
(2)
(3)
(4)
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(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
The objective function (1) maximizes the profit over the operation period for the carsharing
system. The profit is equal to the difference between the revenues earned by users’ making
carsharing trips and the costs yielded by vehicle maintenance and relocation.
Constraints (2) ensures that the number of vehicles at node at the initial time step is equal
to its capacity. Constraints (3) guarantees that the number vehicles at node at time will not
exceed its capacity. Constraints (4) is the flow conservation constraint. It specifies that the
number of vehicles available at node at time equals to the number of vehicles available at
the previous time step minus all the outbound relocation, one-way and round trips at the
previous time step, and plus all the inbound relocation, one-way and round trips at the current
time step. Constraints (5) determines that the number of one-way trips, round trips and
relocation trips could not exceed the number of vehicles available at each vehicle pod.
Constraints (6) and (7) ensure that the number of one-way and round trips are less than or equal
to their travel demand. The travel demand for one-way and round-trip carsharing trips are
calculated by multiplying the total travel demand by the probabilities that they are chosen by
the travelers.
Constraints (8) to (12) calculate the probabilities of choosing one-way and round-trip
carsharing trips based on the MNL model proposed by Catalano et al. (2008). Their model
includes travel time, travel cost, parking time, and the number of household cars as the
parameters that influence traveler’s mode choice. These parameters affect user’s utility linearly.
However, their model does not consider the impact of carsharing vehicle availability on
traveler’s choice. Vehicle availability will affect the accessibility of carsharing services, and
further affect users’ demand. If only a few vehicles are parked in the carsharing vehicle pods,
users may think it is difficult to make a successful booking. In contrast, having more vehicles
parked in the pods will make the users believe they have a higher chance to use this travel mode.
As a result, the utility they perceive to obtain from carsharing will increase. As such, the number
of vehicles available has a positive impact on users’ demand for carsharing. To accommodate
this impact, we add a positive linear term to the utility functions of carsharing. More specifically,
we introduce the number of vehicles available , and its corresponding coefficient to the
carsharing utility functions. It should be noted that the value of is unknown owing to the lack
of data. In order to test the impact of vehicle availability on carsharing demand, a sensitivity
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analysis on is undertaken in Section 5.
Compared to the previous model proposed by Jian et al. (2016), the major difference of
this model lies in Constraints (6) to (12). These constraints incorporate the impact of vehicle
availability on users’ demand. Since the decision variable is involved in the MNL model,
these constraints are nonlinear. The next section will describe the linearization method of the
nonlinear model.
3 Linearization
Constraints (6) to (9) are not linear because the decision variable has a nonlinear impact on
the probabilities of choosing one-way (
) and round trips (
). To linearize these
constraints, the main idea is to decompose
and
by introducing a binary variable
, , and two parameters
and
,
.
represents the probability that a user travels from node to chooses RT
among RT, OW and TS when the number of vehicles available at node is equal to . Similarly,
denotes the probability that a user travels from node to chooses OW among RT, OW
and TS when the number of vehicles available at node is equal to . Here, is a parameter
changing from 0 to the capacity of the corresponding pod (). In this way, the equations of
these two parameters involve the parameter instead of incorporating with the variable .
Constraints (6) to (12) can be rewritten as:
(13)
(14)
(15)
(16)
(17)
Binary variable is introduced to decide if is equal to . We let , if
; otherwise . Then,
is replaced by
, and
is replaced
by
. By doing so, we make sure that
is equal to
and
is
equal to
only when is equal to . Constraints (6) and (7) can then be replaced by the
following linear constraints:
(18)
(19)
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(20)
(21)
This method linearizes the demand constraints by decomposing the decision variable into
the product of a binary variable and a parameter. The decomposition method enumerates several
states of the discrete choice model because of the discrete nature of the number of available
vehicles. It can be generally applied with good rigor in this type of problems that use a time-
space network.
4 Sydney Case Study
The case study is tested on GoGet Sydney carsharing network, the same as that which was used
in the previous optimization study (Jian et al., 2016). GoGet is currently offering round-trip
carsharing services to the users and planning to add one-way trip services to its existing
carsharing system. At the time of data collection, there were 37 frequently-used vehicle pods
located around Sydney urban area containing 198 vehicles in total. We assume that the locations
and capacities of these current vehicle pods remain unchanged after the introduction of one-
way trip services. The study focuses on the AM peak operation period (between 7:00 and 10:00).
To reduce calculation time, the three-hour period is expressed in 13 time steps with 15 minutes
per time step.
The data required to apply the model are: the potential origin-destination (OD) travel
demand matrix, the trip travel times, the parking times, and the number of household cars for
each user. We use Sydney AM peak OD travel demand matrix obtained from EMME, a travel
demand forecasting software (Inrosoftware.com, 2016), to determine the total travel demand
for each vehicle pod at each time step. Sydney AM peak OD travel demand matrix contains the
OD demand between all travel zones within Sydney. We use ArcGIS 10.3 to associate GoGet
vehicle pods to their corresponding travel zones. The total number of trips generated by all
travelers during the AM peak in GoGet operating area is 80930. However, only a proportion of
these trips can be carsharing trips. This is because among all travelers, only GoGet users can
choose to use GoGet carsharing services. Therefore, for each OD pair, we consider a proportion
of the total OD travel demand to be the potential carsharing travel demand. In this way, the total
travel demand from vehicle pod to is equal to a certain proportion of the travel demand from
pod ’s corresponding travel zone to pod ’s corresponding travel zone. The demand multiplied
by a proportion can be a non-integer. To ensure the values of trip demands are integers, we
round the proportioned demands to the nearest integers. Three demand scenarios with
proportions equal to 0.1, 0.2, and 0.45 are tested in this study. The total travel demands with
these three proportions are equal to 3133, 10356, and 28237. These three scenarios represent
low-demand, medium-demand and high-demand cases, respectively.
The trip travel times are computed by dividing the OD route distance by the average
driving speed. The OD route distance is calculated using ArcGIS 10.3 Network Analyst
geoprocessing tool. The average driving speed is assumed to be 30 km/h. This is because GoGet
vehicle pods are located in urban areas with high traffic flows and low traveling speeds.
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Furthermore, the study period is the AM peak which has low average traveling speed. The travel
times of one-way and relocation trips are equal to the travel time of corresponding OD pairs.
The travel time of round trip is the sum of the travel time from the origin to the destination, the
time the user spends at the destination, and the travel time from the destination back to the
origin. The time users wait at the destination is assumed to follow a normal distribution with
the mean equal to 1 time step and the variance equal to 1. Both the travel time and the waiting
time are in number of time steps. The parking time of each trip is assumed to be 0 to simplify
the problem. This is reasonable since carsharing vehicles have dedicated parking spaces, so the
time of finding parking spaces is negligible. The number of household cars for each user is also
assumed to be 0 as the majority of carsharing users do not own a car. The rest of the parameters
are set as follows:
(price rate per time step for round trips): AU$2 per 15 min (AU$1=US$0.74
[14/06/2016]), which is the average rate of GoGet trips.
(cost of maintaining a vehicle per time step driven): AU$0.5 per 15 min.
(cost of relocating a vehicle per time step driven): AU$1.5 per 15 min, which is the
average hourly salary in Sydney.
The values of the coefficients involved in the utility functions formulated are set based on
the MNL model proposed by Catalano et al. (2008). In their model, the unit of time is minute,
whereas the unit in this study is 15 minutes. We multiply the coefficients for travel time and
parking time in their model by 15 to make the model applicable to this study. Furthermore,
in their model, the unit of travel cost is Euro, while the unit of travel cost in this study is
Australian dollar. To make the model applicable, we multiply the coefficient for travel cost
by 0.72 (AU$1= €0.72 [25/03/2017]). The values of the coefficients in the MNL model are set
as follows:
(Coefficient for travel time): -0.3908
(Coefficient for travel cost): -0.2082
(Coefficient for parking time): -1.6155
(Coefficient for number of household cars for each user): -2.6054
(Constant for carsharing): 1.4833
(Constant for other transport services): 1.1489
The price rate of one-way trip and the coefficient of vehicle availability are
considered as sensitivity analysis parameters to study the impacts of one-way trip price and
vehicle availability on system profit, and to understand the interdependent relationship between
demand and supply in carsharing systems. Vehicle pod capacity is also tested in the
sensitivity analysis to study how supply and demand interacts under scenarios with different
capacities.
5 Sensitivity analysis and results discussions
In the optimization approach proposed by Jian et al. (2016), a sensitivity analysis has been
undertaken on one-way trip price and vehicle pod capacity. One-way trip price determines users’
travel demand, while vehicle pod capacity represents the supply of carsharing vehicle fleet. The
findings give insights to how these two factors influence carsharing system profit and implied
the impact of the interdependence between demand and supply on the profit. The sensitivity
analysis in this study aims to investigate how carsharing demand interacts with its supply, and
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how this interaction affects the system profit.
As mentioned in Section 2, we use vehicle availability to link demand and supply. Vehicle
availability represents the supply of vehicles in each pod. On one hand, vehicle availability
determines carsharing demand; on the other hand, it is affected by demand. Therefore, to
understand how the interdependence between supply and demand influences the profit, we can
investigate the impact of vehicle availability on the profit. We introduce the number of vehicles
available and its corresponding coefficient to Catalano et al. (2008)’s model to
accommodate the impact of carsharing vehicle availability on traveler’s choice. The
significance of the impact is determined by the value of . In fact, the ratio of the vehicle
availability coefficient over the absolute value of the travel cost coefficient can be
defined as traveler’s value of vehicle availability (VoVA). For instance, when is 0.1, and
is -0.2, the ratio of over is equal to 0.5, then traveler’s VoVA is AU$0.5 per vehicle. This
indicates a traveler would perceive to save AU$0.5 for every one vehicle available in the vehicle
pod. As VoVA increases, trip cost disutility can be compensated by vehicle availability. As a
result, travelers’ demand become less sensitive to one-way trip price and more elastic to vehicle
availability. In essence, the VoVA dictates the importance of supply on users’ demand. Thus,
the study on the interdependence between supply and demand is interchangeable with the
evaluation of VoVA. Since the exact VoVA is unknown, a sensitivity analysis on VoVA is
conducted in this study. We test six different VoVA, i.e. 0, 0.2, 0.4, 0.6, 0.8 and 1.
In addition, three demand scenarios (low-, medium-, and high-demand scenarios) created
in Section 4 are tested in the sensitivity analysis. For each demand scenario, three capacity
conditions are generated by increasing the capacity from the original capacity to 1.5 and 2 times
higher. Ten one-way trip prices are considered with changing from 3 to 12 with a
discretization step of 1.
Since the waiting time that each round-trip user spends at the destination was randomly
generated, we solved 5 instances for each scenario to achieve statistical convergence. The
model was solved by the commercial package CPLEX (IBM-ILOG, 2016), implemented on an
Intel (R) Core (TM) i7-6600U processor @2.60 GHz 2.81 GHz, 12.0 GB RAM computer with
a Windows 10 64-bit operating system. 2700 instances were solved in total. The average
running time is 24.5 seconds and the standard deviation of the running time is 17.4 seconds.
Every instance was solved to optimality. We calculate the coefficient of variation (CV)
regarding the optimal profit for each instance, and the average CV across all instances is 0.014,
which shows the achievement of statistical convergence.
5.1 Impacts of one-way trip price and vehicle availability on system profit in different
demand scenarios
Three demand scenarios are tested in this study to evaluate how system profit reacts to one-way
trip price and vehicle availability under different travel demand situations. The results of how
profits change over and VoVA with the original capacity (198 vehicles) are illustrated in
Figures 1, 2, and 3.
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Figure 1 System profit over and VoVA when capacity = 198 in low-demand scenario
Figure 2 System profit over and VoVA when capacity = 198 in medium-demand
scenario
Figure 3 System profit over and VoVA when capacity = 198 in high-demand scenario
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Focusing on one-way trip price ( ), it has a significant impact on system profit, but the
importance of varies across the three demand scenarios. According to Figures 1 to 3, the
profit in the low-demand scenario changes more gently over than in the medium- and high-
demand scenarios. This result illustrates that the marginal change in profit with a marginal
increase of becomes larger as the total travel demand increases. This can be explained by
two reasons. First, the system profit is the product of the total number of trips and the unit profit
earned per trip. In scenarios with higher demand, a small change in one-way trip price will lead
to a larger change in the total number of trips accepted. This is consistent with the observations
of the number of trips in Figure 4: the changing pattern of the number of trips over in the
low-demand scenario is smoother than the patterns observed in higher demand scenarios.
Second, when the total travel demand is very low, users’ demand for one-way trips will also be
low. A possible consequence can be that a small increase in will lead the share of one-way
trips to be 0. As shown Figure 4, when VoVA is 0, the number of one-way trips in the low-
demand scenario falls to be nearly 0 when increases to 6. In contrast, in the medium-demand
scenario, the number of one-way trips reduces to be 0 when reaches 12. Therefore, the
changing pattern of system profit over is dependent on the OD travel demand. The operators
should consider the travel demand when determining the optimal one-way trip.
Focusing on the impact of VoVA, Figures 1 to 3 illustrate that system profit becomes less
sensitive to one-way trip price as VoVA increases. In the low-demand scenario as presented in
Figure 1, the profit reaches maximum when is between AU$3 to 5 per 15 minutes with VoVA
equal to 0, whereas the maximal profit is obtained when is larger than AU$6 when VoVA
increases to 1. Compared to the case when there is no VoVA, the maximal profit occurs at a
higher value of one-way trip price. Similar conclusions can be drawn from the medium-demand
and high-demand scenarios. In the medium-demand scenario, the maximal profit is observed at
around AU$5 when VoVA is equal to 0, while at around AU$8 when VoVA is equal to 1.
In the high-demand scenario, the optimal increases from AU$8 to AU$11 when VoVA
increases from 0 to 1. The higher the VoVA, the higher the optimal one-way trip price occurs.
This change in the importance of one-way trip price on profit can be explained by the
introduction of vehicle availability in the utility functions (Constraints (10) and (11)). In the
utility functions, one-way trip price has a negative coefficient and vehicle availability has a
positive coefficient. As VoVA increases, the positive impact of vehicle availability on demand
can eventually offset the negative impact caused by trip cost. As a result, the demand will
decrease less sharply with increasing if VoVA is larger.
This explanation is consistent with the profile of the number of trips presented in Figure 4.
The number of one-way trips increases as VoVA increases. For example, in the medium-demand
scenario, with no VoVA, the number of one-way trips falls from 700 to 0 when increases
from 3 to 12; while with VoVA equal to 1, the number of one-way trips decrease from 1000 to
300. On average, 300 more one-way trips are generated for each price scenario. Since the trend
of system profit over is dependent on travel demand, VoVA will consequently affect the
changing pattern of profit over trip price by changing users’ travel demand.
14
Figure 4 The number of trips over and VoVA in 3 demand scenarios when capacity = 198
5.2 Impacts of capacity on system profit
We next present the results of the sensitivity analysis on capacity for the medium-demand
scenario (total travel demand = 10356) to evaluate the impact of vehicle pod capacity on system
profit. The results are shown in Figure 5.
Three major findings can be concluded from the results. First, the trends of profit over
and VoVA concluded in Section 5.1 are also observed when the capacity inflates. In fact, the
trends of profit and the number of trips at higher capacities are the same as the original capacity,
only the values of profit and the number of trips increase.
15
Figure 5 System profits and the number of trips over and VoVA for different capacities in the medium-demand scenario
16
Second, capacity always has a positive impact on profit. This can be explained by two reasons.
First, higher capacity can guarantee more vehicles available at each vehicle pod. Since users’
demand is positively elastic to vehicle availability, higher capacity can lead to larger market
share for carsharing services. Second, the optimization model proposed in this study rejects
some trip requests due to capacity limit. When the capacity is increased, fewer trip requests will
be rejected, which will yield the increase in system profit. Figure 6 compares the changes in
rejection rates for low- and high-demand scenarios. The average trip rejection rates are reduced
by 0.13% to 0.2% when capacity is inflated from 198 to 396 vehicles.
Figure 6 Rejection rates of low- and high-demand scenarios with different capacities
The third finding is that the importance of capacity on system profit varies across different
VoVA. As shown in Figure 5, the marginal profit gained by a marginal capacity increase is
dependent on carsharing trip demand. When VoVA is equal to 0, the profit increases around
AU$500 with the capacity inflating from 198 to 396. Whereas when VoVA is equal to 1, the
demand for carsharing trips is much larger, and the increase in profit is nearly AU$2500 with
the same capacity inflation. Comparing these two cases, the marginal profit earned by 1 unit
inflation in capacity is increased by 5 times when carsharing trip demand is enlarged by 2 times.
This finding suggests the operators to evaluate the potential carsharing demand before making
network expansion strategies. If the potential demand is high, enlarging carsharing fleet size
can bring substantial profits; while if the carsharing market share is small, the investment on
capacity might not be compensated by the marginal profits brought by the capacity
improvement.
Inspired by the third finding, it is worthy of investigating the optimal ratio of total travel
demand over system capacity (denoted by ), so that the operators can make the most
efficient capacity strategy given local travel demand. In order to solve this optimal ,
we plot how system profit changes over in Figure 7. The optimal that
leads to the maximum profit falls between 5 and 6. This result indicates that the most profit-
efficient capacity planning strategy is to deploy 1 vehicle into the network for every 5 to 6
carsharing users.
17
Figure 7 System profit and number of trips over demand/capacity ratio
5.3 Sensitivity analysis on spare parking spaces
In the optimization model proposed in Section 3, Constraints (2) sets the number of vehicles at
each pod to be equal to the pod capacity, which means there is no spare parking space for each
vehicle pod at the initial time step. If a vehicle pod only has trip attraction, but no trip production,
the operators need to relocate the vehicles parked in that pod to other pods with free parking
spaces. As a result, extra relocation costs will be induced to solve this gridlock situation. Adding
spare parking spaces to each vehicle pod can circumvent such situation and reduce relocation
costs, but the setting up costs of extra parking spaces might not be compensated by the saved
relocation costs.
In order to quantify the profit of adding spare parking spaces, we undertake a sensitivity
analysis on spare parking spaces. We add 1 to 5 parking spaces to each vehicle pod and calculate
the increase in profits with regards to each demand and capacity scenario. The sensitivity
analysis is undertaken with one-way trip price equal to AU$6 and VoVA equal to 0. Figure 8
presents the patterns of the increase in profit over the number of added parking spaces in the
three demand and capacity scenarios.
Three major findings can be withdrawn from Figure 8: 1) Reviewing each capacity
condition, the profit improvement by adding more parking spaces is more substantial in high-
demand scenario than low-demand scenario. This is because when demand is low, a small
number of relocation trips are required. The added parking spaces will only reduce minor
system costs, as a result the profit will experience a marginal increase. 2) Reviewing each
demand scenario, smaller capacity experiences higher increase in profit. The reason is that when
total demand is fixed, larger capacity will attract more carsharing trips, and fewer gridlock
situations will occur. Therefore, adding spare parking spaces will bring less significant
influence on large-capacity network. 3) The marginal gain in profit via adding one parking
space is approaching 0 as the number of spare parking spaces increases. This is because as more
parking spaces added, fewer gridlock situations will occur. Consequently, less profit
improvement will be yielded, and eventually the marginal profit will be 0 as the gridlock
18
situations disappear.
Figure 8 System profit over demand/capacity ratio and parking space relaxation
Concluded from the sensitivity analysis on spare parking spaces, demand is still an important
factor to consider when the operators are determining whether to add spare parking spaces. If
the demand is low, adding extra parking spaces might not be profitable. In contrast, if the
demand is high, adding a few more spaces can be fairly profitable, but the number of spare
parking spaces needs to be decided based on current vehicle pod capacity, potential demand,
and parking space opening costs.
5.4 Computational performance in larger transport networks
In order to evaluate the computational performance of the proposed model in larger networks,
we have undertaken the following numerical experiments: we have considered four transport
networks: Anaheim, Berlin Mitte Prenzlauerberg Friedrichshain Center (MPF), Barcelona,
and Winnipeg, all of which are freely available on the repository
https://github.com/bstabler/TransportationNetworks (Transportation Networks for Research
Core Team). The transport network data includes zone to zone travel demand (trip table) as well
as network topology attributes (nodes and links). To create carsharing instances based on these
transport networks, we have made the following assumptions:
1) We assume there is a carsharing station in each zone of the network. For example, there
are 147 zones in Winnipeg, then based on this assumption, there are also 147 carsharing stations
in this network. The total Origin-Destination (OD) demand (including carsharing demand and
other transport mode demand) of a carsharing station to station pair is set to be equal to the OD
demand of the corresponding zone to zone pair. The capacity of a carsharing station, i.e. the
number of vehicles parked in a station, is randomly set to be an integer between 1 and 20.
2) The network data only provides static OD demand, which is not time-dependent. Since
our optimization model works with dynamic demands, we randomly distribute the total demand
of each OD pair across 12 time intervals of 15 minutes, representing a period of 3h (we use the
same number of time intervals as in the Sydney-based case study for comparison purposes).
3) We use link free-flow travel times to calculate OD travel time and divide the travel time
by 15 minutes to scale it to time steps. The OD pairs free-flow travel times are calculated using
Dijkstra's algorithm.
4) The waiting time of round trips are randomly set to be 0 or 1. The rest of the parameters,
such as round-trip price, relocation cost and maintenance cost, are set to be the same as in the
19
Sydney case study.
We then process the original network data of the four additional city networks considered
and solve the instances under the scenario where VoVA is set to be 0.2. Table 1 compares the
computational times of the four cities and Sydney case study.
Table 1 Comparison of computational performance of five transport networks
Transport
Network
Number of
Carsharing
Stations
Total
Demand
Total
Capacity
Computational time (seconds)
Min
Max
Mean
Sydney
36
3133
198
1.06
8.56
2.84
Anaheim
38
104142
407
2.13
2.83
2.55
Berlin MPF
98
19144
987
10.45
19.84
15.23
Barcelona
110
180487
1222
20.18
54.03
41.12
Winnipeg
147
64737
1484
140.59
223.12
172.17
As listed in Table 1, Anaheim has similar number of stations as Sydney, but much higher
demand. The average running time of Anaheim instances is close to and even a bit shorter than
Sydney. Berlin MPF has nearly three times more stations than Sydney, and also a higher demand.
The average computational time for Berlin MPF is approximately 5 times that observed when
solving the model for Sydney’ s network. Barcelona and Winnipeg are two large-size networks,
with both having more than 100 stations with over 1200 vehicles. The average computational
times of these two networks increase exponentially compared to that of Sydney’s network.
Concluded from Table 1, the computational time of the proposed model increases as the
network size, demand, and capacity increase. Among them, network size is the dominant factor
that affects the computational time.
Although the running times of larger networks are substantially higher than the Sydney
case study, those running times are still acceptable. As presented in Table 1, the maximum
running time of Winnipeg network is close to 4 minutes, which is much shorter than the
operational periods. Therefore, the computational performance of the proposed model is
acceptable and model is applicable for real-world operational decisions.
6 Conclusions
The potential of attracting a larger market share has urged carsharing operators to introduce
one-way carsharing services to their existing round-trip carsharing systems. However, the
flexibility of one-way carsharing services can cause some logistical problems to the systems
due to potential imbalances in vehicle supply and demand. Previously, researchers have
proposed several solutions to the problem, such as operator-based relocation approach, user-
based relocation method, and the station location solution. A limitation of previous methods is
the ignorance of the interdependence between supply and demand in carsharing systems: the
demand for carsharing trips is influenced by the supply of vehicle fleet, and the demand further
changes the vehicle supply in carsharing vehicle pods. The study proposed by Jian et al. (2016)
considered the interaction between demand and supply when formulating the relocation model,
however, the model did not accommodate vehicle supply as a factor that directly affected users’
travel demand. In this study, we extended the aforementioned model by including vehicle
availability as a parameter determining users’ mode choice within the discrete choice model.
20
Vehicle availability linked supply and demand by integrating the relocation optimization model
with the discrete choice model. This led the extended model to be nonlinear. To deal with the
nonlinearity, the study introduced new binary variables and parameters to decompose the
nonlinear terms of the model. Linearizing the constraint of the discrete choice model is the
major contribution of this paper. The proposed linearization method can be applied to linearize
models incorporating discrete choice models as constraints.
The model was tested on the carsharing network of GoGet in the metropolitan area of
Sydney using realistic operation data. The results of the sensitivity analysis reveal that
carsharing system profitability is influenced by a complex group of factors, including trip price,
system capacity, total travel demand, and users’ elasticity to vehicle availability. Among them,
the impact of one-way trip price on profit varies as VoVA changes. When VoVA is small, the
demand is more sensitive to the one-way trip price, and the maximal profit occurs at lower one-
way trip price. As VoVA increases, the impact of one-way trip price on travelers’ demand is no
longer dominant. Instead, the demand becomes more elastic to vehicle availability. Therefore,
increasing one-way trip price will reduce the demand more gently, and the optimal one-way
trip price appears at a larger value. This result is different from the previous sensitivity analysis
which does not accommodate vehicle availability as a parameter in the discrete choice model.
The difference in the profit patterns suggests if supply has a significant impact on demand,
demand is less sensitive to trip price, and higher trip price will generate more profits. With
regards to system capacity, the results suggest that although capacity always has a positive
impact on profit, the marginal profit of increasing one unit capacity is higher with high travel
demand than low travel demand. The optimal demand over capacity ratio falls between 5 to 6.
Furthermore, the sensitivity analysis on additional parking spaces implies that the marginal
profit of adding one unit parking space is also higher when demand is higher. These findings
suggest operators that it is critical to evaluate the potential travel demand before making
network capacity development plans.
The main conclusion we can withdraw from the study is that the interdependence between
demand and supply should be taken into consideration when designing carsharing network size
and setting pricing strategies in one-way carsharing systems. Further study will concentrate on
conducting a stated preference survey to determine the exact coefficient of vehicle availability.
Formulating heuristic algorithms to shorten the computing time and provide scalable solution
methods will also be the focus of the future work.
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