Transportation Research Part B 104 (2017) 17–35

Contents lists available at ScienceDirect

Transportation Research Part B

journal homepage: www.elsevier.com/locate/trb

Determining optimal locations for charging stations of electric

car-sharing systems under stochastic demand

Georg Brandstätter, Michael Kahr, Markus Leitner ∗

University of Vienna, Faculty of Business, Economics and Statistics, Department of Statistics and Operations Research, Vienna, Austria

a r t i c l e i n f o

Article history:

Received 6 November 2016

Revised 11 June 2017

Accepted 13 June 2017

Available online 27 June 2017

Keywords:

Location analysis

Car-sharing

Electric cars

Time-dependent formulations

Integer linear programming

Stochastic optimization

a b s t r a c t

In this article, we introduce and study a two-stage stochastic optimization problem suit-

able to solve strategic optimization problems of car-sharing systems that utilize electric

cars. By combining the individual advantages of car-sharing and electric vehicles, such

electric car-sharing systems may help to overcome future challenges related to pollution,

congestion, or shortage of fossil fuels. A time-dependent integer linear program and a

heuristic algorithm for solving the considered optimization problem are developed and

tested on real world instances from the city of Vienna, as well as on grid-graph-based in-

stances. An analysis of the influence of different parameters on the overall performance

and managerial insights are given. Results show that the developed exact approach is suit-

able for medium sized instances such as the ones obtained from the inner districts of

Vienna. They also show that the heuristic can be used to tackle very-large-scale instances

that cannot be approached successfully by the integer-programming-based method.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

The expected growth of the human population by two to four billion in the first half of the 21st century will impose

severe challenges to humanity. Some of them are intensified by the increasing trend towards urbanization, especially in de-

veloping countries ( Cohen, 2003 ). Two main sources of such challenges are the expected large increase in demand of energy

and transportation. Without significant changes in the mode of transportation and type of fuels used, increased demands

may lead to shortages of fossil fuels, which are still the dominating sources of energy and are estimated to be exhausted

before 2050 ( Shafiee and Topal, 2009 ). In addition, severely amplified problems with respect to pollution, congestion, noise

and lack of parking space are expected. These challenges might be partly met by the consideration and implementation of

new concepts of transportation such as car-sharing systems, which can reduce the number of circulating cars ( Martin et al.,

2011 ), as well as the total distance traveled by them, see Shaheen et al. (2009) . Hence, such systems are likely to reduce

congestion-related delays and to free up parking space ( Crane et al., 2012 ). A possibility to overcome the rapid exhaustion

of fossil fuels and therefore to decrease the emissions of greenhouse gases is the increased usage of electric-powered ve-

hicles (EVs), premised that the electric power comes from clean energy sources ( Granovskii et al., 2006 ). The fact that the

market-share of EVs was extremely low (0.1%) compared to the number of all passenger vehicles worldwide in 2015 un-

derlines their potential ( International Energy Agency, 2016 ). However, a major disadvantage of EVs is the large amount of

time needed for recharging them compared to the amount of time needed to refuel conventional vehicles, and the lack of

∗ Corresponding author.

E-mail addresses: georg.brandstaetter@univie.ac.at (G. Brandstätter), m.kahr@univie.ac.at (M. Kahr), markus.leitner@univie.ac.at (M. Leitner).

http://dx.doi.org/10.1016/j.trb.2017.06.009

0191-2615/© 2017 Elsevier Ltd. All rights reserved.

18 G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35

(private) charging stations in urban areas, which impedes the growth of privately held EVs. A successful implementation of

car-sharing systems that utilize EVs, i.e., electric car-sharing systems , may help overcome the aforementioned environmental

challenges as they combine the advantages of car-sharing and electric vehicles.

Operators or cities supporting electric car-sharing systems must, however, address difficult strategic questions before

possibly opening their business. Besides selecting an appropriate mode and area of operation, these particularly include the

question of where to build charging stations at which cars can be recharged while not used by customers. Additional aspects

that shall be considered include the number of purchased cars and their (initial) distribution over the operational area in

order to best meet customer demands. To this end, several car-sharing concepts are known, which we summarize in the

following paragraph where we also address the question of their applicability to electric car-sharing systems.

Car-sharing concepts. One main classification of car-sharing systems describes whether users are only allowed to pick up

and drop off cars at designated stations (i.e., station-based systems) or whether trips can, in principle, be started and ended

at any free parking spot in the operational area (i.e., free-floating systems). Both variants can be implemented either as a

traditional system in which pre-bookings for predefined time periods are required, which include the specification of pick-up

and drop-off locations, or as ad-hoc systems that do not require pre-bookings. Note that ad-hoc systems may nevertheless

include the possibility of reserving a vehicle for a short period of time before starting a trip, thus avoiding the case where

an available car is taken by another customer in the meantime. Finally, while most systems include the possibility of one-

way trips , others are restricted to round-trips for which the pick-up and drop-off location needs to coincide, see, e.g., Boyaci

et al. (2015) .

We observe that in urban settings, users will typically ask for a maximum amount of flexibility (i.e., prefer ad-hoc free-

floating systems) while an operator might prefer a system that is easier to manage (i.e., a station-based system with pre-

bookings). Station-based systems have additional advantages for operators in the context of electric car-sharing. Each station

can be equipped with charging slots at which idle cars can be recharged. Another benefit is that the pre-booking process

can help prevent both shortages and excess supply of cars in the stations by planning the relocation of vehicles in advance.

Users can also benefit from the station-based concept because operators can ensure that a free vehicle is available at the

desired origin, and that a free parking slot is available at the desired destination, due to the pre-booking procedure (an

operator can, e.g., lock a pre-booked car at the origin for a given amount of time, and reserve a free parking slot at the

desired destination). Along the same lines, several disadvantages of free-floating electric car-sharing systems are observed.

Frequent recharging of empty vehicles and the poor predictability of shortages and excess supply in specific areas require

the use of powerful user- or operator-based relocation strategies, see, e.g. Barth et al. (2004) and Kek et al. (2006) .

Contribution and outline. In this article, we introduce a combinatorial optimization problem targeting the strategic plan-

ning process of electric car-sharing systems in order to appropriately support decision makers. Given a stochastic demand

forecast, its aim is to identify optimal locations for charging stations and an associated number of required EVs in order

maximize the expected profit, obtained from accepting trips in a predefined planning period. After a brief literature review

in Section 2 , the new problem is described in detail and formally defined in Section 3 . A two-stage stochastic integer linear

program (ILP) for the considered problem in its deterministic equivalent is introduced in Section 4 , while Section 5 details

a heuristic solution approach. The latter is used to either compute an initial solution when solving a benchmark instance

to optimality with our ILP formulation, or as a stand-alone heuristic to compute good, but typically not optimal solutions

to large-scale instances. Computational results obtained from solving grid-graph instances are discussed in Section 6 , where

we focus on a more general performance analysis and the influence of different input parameters. In Section 7 , we dis-

cuss results and findings obtained from applying the developed methods to real world instances from Vienna and provide

managerial insights. Conclusions are drawn in Section 8 , where we also point out directions for future research. Finally,

Appendix A contains a list of the most important sets, variables and parameters used throughout this article.

2. Previous and related work

Though an increased interest in electric car-sharing systems can be observed recently, the existing scientific literature

related to optimization problems in such systems is still relatively scarce. Several articles do, however, address the case

of privately owned or commercial electric vehicles. A more in-depth overview of the current state of the art regarding

optimization problems arising in the context of electric car-sharing, as well as the aforementioned related areas, is given by

Brandstätter et al. (2016) in their recent survey on the topic.

We first summarize relevant literature related to the placement of public charging stations for privately owned electric

vehicles, before turning our attention to literature on (electric) car-sharing. Worley et al. (2012) developed an integer linear

program that determines optimal locations for charging stations for fleet owners and simultaneously suggested routes for

the vehicles. Instead of maximizing profits, the given model minimizes the total costs. A mixed integer linear program that

minimizes the total access costs to charging stations based on walking distances was introduced by Chen et al. (2013) .

They used data from over 30 0 0 0 records of personal trips. Nie and Ghamami (2013) investigated how to select the battery

size and the capacity in terms of the number of charging stations and charging power needed, in order to meet a given

level of service. Their objective was to minimize the social costs. A genetic algorithm to find (sub)optimal locations for

public charging stations for EVs was developed by Dong et al. (2014) , who also provided a case study based on multiday

G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35 19

GPS-based travel survey data. They showed that the traveled miles and the number of trips of EVs could be significantly

increased by installing public charging stations at popular destinations, with reasonable infrastructure investment. Another

related study has been recently performed by Faridimehr et al. (2017) . They proposed a two-stage stochastic ILP (tackled

via sample-average approximation) and a heuristic algorithm with the goal of maximizing the demand of private vehicles

(trips) that can be covered through the established charging stations. Data uncertainty stemming from various sources (e.g.,

arrival and departure times) is considered. Results from a case study in Detroit midtown (Michigan, US) are reported.

Barth and Todd (1999) proposed a queuing-based discrete-event simulation model for electric car-sharing systems with

the objective to analyze operational issues. They state that the most effective number of vehicles is in the range of 3–6 per

100 trips based on a 24 h day. If the objective is to minimize the number of relocations, they suggest that the number of

vehicles should be approximately 18–24 per 100 trips. Their model shows that the steadiness of the operation of car-sharing

systems is most sensitive to the car-to-trip ratio. Cepolina and Farina (2012) studied a multi-station electric car-sharing sys-

tem using real-word data from Genoa. They provided an optimization model to determine the dimension of the fleet and

the distribution over the stations with the objective of minimizing the overall costs, i.e., costs of the transportation system

and the users costs, whereby the latter depend on waiting times. Recharging of the vehicles is assumed to happen at idle

times. An existing one-way, pre-booked electric car-sharing system in Kyoto was investigated by Nakayama et al. (2002) .

They developed a simulation model with the objective of maximizing the check-outs of the EVs, with the number of ve-

hicles, capacities of the stations and the number of users as decision variables. Results suggest that the optimal number

of vehicles is about half of the total amount of parking slots. Boyaci et al. (2015) presented a multi-objective integer lin-

ear program that optimizes strategic decisions related to the placement of stations and the fleet size, while also allowing

for the operator-based relocation of cars throughout the day. However, while our approach enforces necessary recharging

stops based on the actually fulfilled demand, theirs relies on it being known a priori. Li et al. (2016) approached the prob-

lem of minimizing the overall system costs of one-way car-sharing systems that utilize EVs, while considering stochasticity

of demands (i.e., trips). In order to overcome computational challenges, they proposed a continuum approximation model

that decomposes the studied area into a number of small neighborhoods such that each can be approximated by an infi-

nite homogeneous hyperplane. They showed that the solutions of this method are able to approximate those of its discrete

counterpart efficiently and with high accuracy, even for large-scale heterogeneous problems. Moreover, they performed a

case study using the transportation network of Sioux-Falls city (North Dakota, US) and drew several managerial insights. Re-

cently, Brandstätter et al. (2016) introduced and studied an optimization problem targeting the planning of charging stations

of electric car-sharing systems. The authors present several ILP formulations and two heuristic methods, develop appropriate

solution algorithms, and compare them empirically with respect to their performance. In contrast to the present article, they

consider the case of deterministic demands and also focus on the question how to best model a detailed battery tracking

per car by ILP formulations.

Finding the best locations and sizes of car-sharing stations for traditional vehicles has been investigated by Rickenberg

et al. (2013) . They suggested a mixed integer linear program with the objective of minimizing the total costs while satisfying

customer demand and preferences. Furthermore, they assumed a stochastic demand modeled by a normal distribution.

In conclusion, we observe that most research including stochastic demand in the context of car-sharing focuses on vehicle

relocation in order to prevent shortages and excess supply in parts of the service area, see, e.g., Barth et al. (2004) , Bruglieri

et al. (2014) and Kek et al. (2006) . Notice that relocation is part of decision-making at the operational level. Remarkably,

there is little research on determining optimal locations for car-sharing stations with respect to stochastic demand, which

happens at the strategic planning level.

A problem related to optimal placement of charging stations in urban areas deals with the placement of such stations

between cities. In that context, trips undertaken by customers are usually too long to be feasible on a single battery charge,

which necessitates en-route recharging. Models and algorithms for solving these intercity charging station location problems

are described by Arslan and Kara ̧s an (2016) , as well as by Kuby and Lim (2005) and Capar et al. (2013) who consider the

analogous problem of placing refueling stations for alternative-fuel vehicles. In these articles, the authors seek to cover the

recharging or refueling demand of trips by placing appropriate stations along them instead of at their start or end, which

is opposite to the requirements of an urban car-sharing system like the one considered in the present article. Furthermore,

none of these articles considers the use-case of car-sharing and the associated modeling of available cars at individual

stations.

The problem of finding locations for bike-sharing stations is closely related to that of finding such locations for charging

stations in a station-based one-way electric car-sharing system. This problem, in conjunction with that of optimizing the

network structure of bike paths between the stations and the corresponding routing of bicycles, is studied by, among others,

Lin and Yang (2011) and Lin et al. (2013) . Naturally, the authors do not consider imposing recharging stops, as these are

not necessary for bicycles. Consequently, their models cannot be used directly for the case of electric car-sharing as the

latter might run out of battery during trips if one would simply adopt the obtained solutions. Furthermore, the authors of

both articles consider only yearly travel demands between points in their models, which cannot account for hourly, daily or

seasonal differences in demand. Station capacities are assigned in such a way as to cover the daily demands (that is assumed

to follow a certain probability distribution) with a certain probability. Even algorithms that incorporate electric bicycles, like

the one proposed by Martinez et al. (2012) , do not consider provisioning for such recharging breaks, which have been an

important consideration in other optimization problems dealing with electric vehicles, such as the electric vehicle-routing

problem with time windows and recharging stations proposed by Schneider et al. (2014) . The authors also assume that all

20 G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35

demands travel along the shortest paths from origins to destinations, the lengths of which also determine duration and

profit of each trip. Such a demand model does not allow for the incorporation of round-trips or trips with multiple stops

or detours (which might, for instance, occur when customers use car-sharing for shopping trips), or for the implementation

of more flexible pricing schemes with discounts for longer rentals (which are used by major car-sharing providers). None of

the aforementioned articles on bike-sharing station placement perform a detailed computational study where the presented

algorithm’s performance and its dependence on various instance characteristics is analyzed. Another frequently considered

optimization problem related to bicycle sharing is concerned with re-balancing the system by relocating bicycles between

stations. Both exact and heuristic algorithms for solving this problem have been proposed, see, e.g., Erdo ̌gan et al. (2014) ;

2015 ) or Di Gaspero et al. (2013) . In these articles, the set of charging stations is, however, given as an input rather than

being part of the optimization process. Also, since these problem settings commonly assume that multiple bicycles can be

relocated at once (such as by a truck transporting them), it remains unclear whether these algorithms can be easily adapted

to the problem of re-balancing electric car-sharing systems where this is not the case.

Finally, we note that the problem studied in this article is related to location-routing problems (LRP) which combine two

fundamental planning tasks in logistics, i.e., determining optimal facility locations and vehicle routing, see, e.g., Drexl and

Schneider (2015) , Nagy and Salhi (2007) and Prodhon and Prins (2014) for recent surveys. Thereby, a well-known result is

that making these types of decisions independently may lead to suboptimal planning ( Salhi and Rand, 1989 ). However, if

we assume that the set of opened facility locations is already fixed and that the capacity constraints are not binding, our

problem is also closely related to a special case of the one-to-one multi-commodity vehicle routing problem with pickups

and deliveries (PDP), also known as the dial-a-ride problem in the context of person transportation in the literature, see,

e.g., Berbeglia et al. (2007) and Parragh et al. (2008) for relatively recent surveys. Hence, the problem under investigation is

related to a combination of the LRP and the PDP, also known as pickup-and-delivery LRP in literature, see, e.g., Karaoglan

et al. (2012) . In the following, we detail the main similarities to and differences between the problem studied in this ar-

ticle and the aforementioned problems. Each trip in our problem corresponds to a commodity to be delivered in the PDP.

However, our objective is the maximization of the expected profits obtained from fulfilling stochastic requests in a given

planning period, whereas the PDP is usually concerned with minimizing the cost of satisfying all requests – thus, our prob-

lem is more similar to a prize-collecting (or team orienteering) PDP variant. Moreover, we restrict time windows for pickup

and delivery to single time points, set the vehicle capacities to one (i.e., each cargo has to be delivered immediately after

its pick-up) and allow locations to be visited multiple times and by multiple vehicles. Notice that, in contrast to the regular

LRP where facility opening decisions are usually only made for potential depots, every location (even those only serving as

start and end point for customer requests) must be explicitly opened in order to be usable by vehicles in our problem. Also,

whereas capacity limits in pickup-and-delivery LRP restrict the amount of commodities served from a particular depot, the

capacity limits constrain the number of simultaneously parked vehicles at each location in our case. Furthermore, we enforce

trip-duration-dependent recharging breaks at each commodity’s destination, and therefore restrict simultaneous pickups and

deliveries performed by a single vehicle. However, we do allow multiple potential start and/or end points for each commod-

ity, since we assume that customers are willing to walk to potential start stations within a given walking distance from

their trip origin, and from potential end stations within that walking distance to their trip destination, respectively. Hence,

trip requests can be fulfilled in multiple ways, whereby exactly one available origin-destination pair has to be selected for

each accepted trip request. Note that this is different from the many-to-many PDP in the sense that each commodity is only

delivered once from any start to any end point in our problem, whereas it must be delivered from every start to every end

point in the many-to-many PDP. In addition, we do not allow vehicles to travel without carrying a commodity, as the latter

would correspond to a car moving without performing a trip – and thus, without a driver – in our setting.

3. Problem definition

As mentioned in the introduction, the present work focuses on decision support at the strategic planning level of elec-

tric car-sharing systems. Thus, the stochastic charging station location problem (SCSLP) introduced in the following aims to

choose a set of charging stations to build and the number of electric cars to purchase in order to maximize the expected

profit during operation, which is earned from accepted customer trips. Thereby, we focus on a one-way, station-based sys-

tem. Note that our model is able to address ad-hoc car-sharing systems as well as those that require pre-bookings. This

observation stems from the fact that strategic planning is performed based on estimated customer demands that may, for

example, be obtained from historical data or surveys. Once stations are built, an operator may decide on a more traditional

variant with required pre-bookings or an ad-hoc system that will be harder to manage.

The formal definition of the SCSLP given below is based on the following assumptions:

• A customer demand forecast is available for different scenarios (e.g. seasons, weekdays) with associated probabilities.

Each scenario is given as a set of estimated trips that account for spatial and temporal aspects of future demand. Each

trip contains information about its source, destination, start- and end-time, which in turn allow us to estimate (bound

from above) its maximum battery consumption and profit contribution. • The mode of planning is conservative in the sense that each selected trip needs to be assigned to an initially fully charged

vehicle. While this assumption is likely to be relaxed in operation, it ensures that small mistakes in the estimation of

battery consumptions (see above) will not have severe impacts in the sense that a car will run out of battery during a

G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35 21

Fig. 1. Two options for routing a trip k from origin o k to destination d k when assuming that stations i 1 , i 2 and i 3 are built (i.e., { i 1 , i 2 , i 3 } ⊆S ′ ) and that

stations i 2 and i 3 are within walking distance of d k (i.e., { i 2 , i 3 } ⊆N ( d k )). (a) Client returns the car at station i 2 ; (b) Client returns the car at station i 3 . Solid

arcs represent the trip taken by the rental car, while dashed arcs indicate walking from / to stations.

trip. To this end, note that in urban areas, trips are usually battery-feasible when started with a fully charged car, see

Duchrow et al. (2012) . This also holds for all trips in the problem instances considered in our computational experiments.

Consequently, we do not consider en-route charging possibly performed by customers, which would induce rather long

waiting times that are unacceptable for intra-city trips. For simplicity, we also assume linear recharging characteristics

(with respect to time). Our approach is, however, flexible enough to consider arbitrary (non-linear) charging functions as

long as the required time to fully recharge a vehicle (from some arbitrary, but known, battery state) can be precomputed.

These assumptions (linear charging and fully recharging a vehicle at each recharging stop) are also in line with the

assumptions made for other optimization problems using electric cars, such as the electric vehicle-routing problem with

time windows and recharging stations, see Schneider et al. (2014) . • The potential locations for charging stations have individual maximum capacities (i.e., maximum numbers of charging

slots) which are subject to local conditions. If a station is built, it is equipped with the maximum number of charging

slots possible. • Customers are willing to walk from their origin to a charging station and from a charging station to their destination, as

long as the associated walking distance (time) does not exceed a given threshold. • Operators acquire a homogeneous fleet of EVs to ease the planning and maintenance effort. • Decision makers do not consider operational activities of the service staff, such as car relocation.

For each instance of SCSLP, let digraph G = (V, A ) with vertex set V and arc set A represent the street network of the

potential operational area. Set S ⊆ V describes the potential locations of charging stations where each i ∈ S has associated

construction (building) costs F i ≥ 0, operating costs ϕi ≥ 0, and a capacity C i ∈ N describing the number of charging slots

that can be built at that station. For each station i ∈ S , its neighborhood N i ⊆ V is the set of nodes that are within walking

distance (time), i.e., from/to which a user is willing to walk to/from that station. Similarly, the set of potential stations

in the neighborhood of a vertex v ∈ V is defined as N(v ) = { i ∈ S | v ∈ N i } . Parameter H ∈ N defines the maximum number

of available cars, each of which has acquisition costs F car ≥ 0, operating costs φ ≥ 0, a battery capacity of B max , and a

recharging rate of ρ , 0 < ρ ≤ B max , per time unit. The available budget for constructing stations and purchasing cars is

given by W ≥ 0.

Each instance further contains a demand forecast that is based on the set T = { 0 , . . . , T max } of discrete time points in

the planning period. Let � be the set of demand scenarios and �ω > 0 be the probability of scenario ω ∈ � such that∑

ω∈ � �ω = 1 . Each scenario ω ∈ � consists of a set of trips K

ω , whereby each trip k ∈ K

ω is given as a tuple ( o k , d k , s k , e k ,

p k , b k ). Thereby, for each trip k ∈ K =

⋃

ω∈ � K

ω , o k ∈ V and d k ∈ V denote the origin and destination of trip k , while s k ∈ T

and e k ∈ T are the associated start and end times. Notice that s k < e k and that we denote by �k = e k − s k the duration of

trip k . Finally, p k > 0 is the profit contribution of trip k representing its revenue reduced by its (estimated) variable costs

while b k , 0 ≤ b k ≤ B max , is its (estimated) battery consumption. As mentioned above, we assume that an operator is able

to compute an upper bound on a trip’s battery consumption based on its duration �k , its origin o k , and its destination d k ,

respectively. We observe that the time required to fully recharge a car after trip k is given as

⌈

b k ρ

⌉

.

The objective of SCSLP is to select a set of charging stations S ′ ⊆S that are built and a number H

′ ≤ H of cars that are pur-

chased such that the expected profit obtained from accepted trips is maximized. Thereby, the total costs of all built stations

and all purchased cars may not exceed the available budget, i.e., ∑

i ∈ S ′ F i + F car H

′ ≤ W . For each scenario ω ∈ �, a trip k ∈K

ω can only be accepted if it is assigned a purchased car h , 1 ≤ h ≤ H

′ , a built start station start ( k ) ∈ N ( o k ) ∩ S ′ in the neigh-

borhood of its origin, and a built end station end ( k ) ∈ N ( d k ) ∩ S ′ in the neighborhood of its destination. Thus, from a users’

perspective, trip k consists of first walking from o k to station start ( k ), driving the assigned car to station end ( k ), and finally

walking from end ( k ) to d k , see Fig. 1 . For each ω ∈ � and each purchased car h , 1 ≤ h ≤ H

′ , let K

′ h (ω) = (k ω

1 , k ω

2 , . . . , k ω

l ) be

the sequence of trips performed with car h in temporal order, i.e., e k ω j

≤ s k ω j+1

, ∀ j ∈ { 1 , 2 , . . . , l − 1 } . To be feasible, each such

sequence must satisfy the following conditions: the end station of a trip must be the start station of the subsequent trip,

22 G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35

Fig. 2. (a) Problem instance with two scenarios ω ∈ � of equal probability, and (b) optimal solution to this instance. Each potential station i ∈ S has a

construction cost F i = 10 0 0 and cars can be purchased at the cost F car = 100 , but there is only a budget W of 3500 available. The trips k ∈ K ω are given in

temporal order such that e k ω j

< s k ω j+1

, and each trip generates a profit contribution of 5. Solid arcs represent the trips of the first scenario and dashed arcs

those of second, respectively. Due to the budget limitation, station i 3 cannot be constructed, and therefore trip k 1 2 must be discarded. Note that only one

car is purchased in the optimal solution.

i.e., end (k ω j ) = start (k ω

j+1 ) , 1 ≤ j < l ; all used start and end stations must be built, i.e.,

⋃ l j=1

({ start (k ω

j ) , end (k ω

j ) }

)⊆ S ′ ; and

the temporal break between any two successive trips must be sufficient to fully recharge the car, i.e., s k ω j+1

≥ e k ω j

+ b k ω j /ρ� ,

1 ≤ j < l . Finally, each solution must also meet the capacity constraints imposed by built stations. Thus, for each scenario

ω ∈ �, the number of cars that are simultaneously at station i ∈ S ′ may not exceed its number of charging slots C i dur-

ing the planning period. To this end, car h is at time t at station start (k ω 1 ) if 0 ≤ t < s k ω

1 , at station start (k ω

j ) , 2 ≤ j ≤ l , if

e k ω j−1

≤ t < s k ω j , and at station end (k ω

l ) if e k ω

l ≤ t ≤ T max . Fig. 2 shows an instance together with a feasible solution and also

indicates the trips realized in the two scenarios considered.

4. Time-dependent integer linear program

In this section, we introduce a two-stage stochastic integer linear programming formulation for the SCSLP in its deter-

ministic equivalent form. Formulation (1)–(6) uses the following two sets of first-stage decision variables: variables y i ∈ {0,

1}, ∀ i ∈ S , indicate whether a station is built or not, while variables z h ∈ {0, 1}, ∀ h ∈ { 1 , 2 , . . . , H} , indicate whether or not

car h is purchased. Finally, variables x k ∈ {0, 1}, ∀ k ∈ K

ω , ∀ ω ∈ �, indicate if a trip k can be accepted in scenario ω. The

latter variables are, however, second-stage decisions, that are included in (1)–(6) in order to define the objective function

(1) . Further second-stage decision variables will be introduced below.

max ∑

ω∈ ��ω

( ∑

k ∈ K ω p k x k

)

−∑

i ∈ S ϕ i y i −

∑

h ∈ H φz h (1)

s . t . ∑

i ∈ S F i y i +

∑

h ∈ H F car z h ≤ W (2)

(x , y , z ) ∈ X

ω ∀ ω ∈ � (3)

x k ∈ { 0 , 1 } ∀ k ∈ K

ω , ∀ ω ∈ � (4)

y i ∈ { 0 , 1 } ∀ i ∈ S (5)

z h ∈ { 0 , 1 } ∀ h ∈ { 1 , 2 , . . . , H} (6)

The objective function (1) maximizes the expected (second-stage) profit contribution of the accepted trips reduced by

the operating costs of the built stations and purchased cars. The budget constraint (2) accounts for the limited budget W .

For each scenario ω ∈ �, abstract constraints (3) are used to state that there must be a way to extend the partial solution

implied by the first stage decisions (stations, cars) and the accepted trips to a feasible solution of the SCSLP. To this end, set

X

ω contains all incidence vectors ( x, y, z ) such that one can find a start station, an end station, and a purchased car to each

accepted trip such that the trip sequence associated to each car must be a feasible route. In addition, the capacity constraints

imposed by the built stations must be met by the union of these routes in each scenario, see the solution description in

G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35 23

Fig. 3. Time-expanded location graph for the first scenario of the instance given in Fig. 2 a. The second-stage solution from Fig. 2 b is indicated by bold

arcs. Stations i 1 , i 2 , i 4 are opened, i.e., S ′ = { i 1 , i 2 , i 4 } , while potential station i 3 remains closed. Consequently, trip k 1 2 , which is indicated by the dotted line,

cannot be performed.

Section 3 for more details. Our ILP formulation for modeling constraints (3) is based on tracking each car’s position at each

point in time and in each scenario by considering the following set of time-expanded location graphs , which are also known

as time-space networks in literature.

Time-expanded location graphs. To enable tracking the position of each car at each time point, we introduce a time-expanded

location graph G

ω = (V ω , A

ω ) for each scenario ω ∈ �. Node set V

ω consists of a source (root) node r ω , a sink node s ω , and

one node i t for each station i ∈ S and each considered time point t ∈ { 0 , 1 , . . . T max } . Arc set A

ω is the union of waiting arc

A

ω W

= { (i t , i t+1 ) | i ∈ S, t ∈ { 0 , 1 , . . . , T max − 1 } , travel arcs A

ω T

=

⋃

k ∈ K ω A

ω T (k ) , initial allocation arcs A

ω I

= { (r ω , i 0 ) | i ∈ S} , and

final collection arcs A

ω C

= { (i T max , s ω ) | i ∈ S} . Thereby, A

ω T (k ) = { (i s k , j e k ) | i ∈ N(o k ) , j ∈ N(d k ) } is the set of trip arcs corre-

sponding to trip k ∈ K

ω . Note that a time-expanded location graph contains parallel travel arcs if two or more trips in the

same scenario have identical start and end times and if, additionally, their sets of potential start and end stations, respec-

tively, overlap. Waiting arcs will be used to represent parked cars whose batteries are being charged in the corresponding

time interval, while travel arcs will model performed trips (with appropriate battery usage). Allocation arcs used in a solu-

tion will be interpreted as initially placing cars at the corresponding station, while final collection arcs will turn out to be

necessary for enforcing the capacity constraints at the end of the planning period. Notice that the graph does not contain

parallel waiting, allocation or collection arcs, since each arc will be linked to each available car. An example time-expanded

location graph and a solution corresponding to scenario one of Fig. 2 is given in Fig. 3 .

We define second-stage flow variables f h a ∈ { 0 , 1 } , ∀ h ∈ { 1 , 2 , . . . , H} , ∀ a = (i t , j t ′ ) ∈ A

ω , ∀ ω ∈ �, to indicate whether car

h travels from station i at time t to station j at time t ′ . Note that traveling along a waiting arc (i t , i t ′ ) corresponds to parking

(and recharging) a car at station i from time t until time t ′ . Additionally, we also use variables x h k

∈ { 0 , 1 } , ∀ h ∈ { 1 , 2 , . . . , H} ,∀ ω ∈ �, ∀ k ∈ K

ω , that will be equal to one if and only if an accepted trip k of scenario ω will be realized by purchased

car h . Using these and all previously defined variables, abstract constraints (3) are realized by (7) –(16) . In this formulation,

for each scenario ω ∈ � and each node u ∈ V

ω , we also use notations δ+ (u ) = { (u, v ) ∈ A

ω } and δ−(u ) = { (v , u ) ∈ A

ω } to

refer to the set of outgoing and incoming arcs of a node u , respectively. For a subset of arcs A

′ ⊂ A

ω , we also use notation

f h [ A

′ ] =

∑

a ∈ A ′ f h a .

H ∑

h =1

x h k = x k ∀ ω ∈ �, k ∈ K

ω (7)

x h k ≤ z h ∀ h ∈ { 1 , 2 , . . . , H} , ∀ ω ∈ �, ∀ k ∈ K

ω (8)

H ∑

h =1

∑

a ∈ δ+ (i t ) ∩ (A ω W

∪ A ω C )

f h a ≤ C i y i ∀ ω ∈ �, ∀ i t ∈ V

ω \ { r ω , s ω } (9)

f h [ δ−(i t )] ≤ y i ∀ h ∈ { 1 , 2 , . . . , H} , ∀ ω ∈ �, ∀ i t ∈ V

ω \ { r ω , s ω } (10)

f h [ A

ω I ] = z h ∀ h ∈ { 1 , 2 , . . . , H} , ∀ ω ∈ � (11)

f h [ δ−(i t )] = f h [ δ+ (i t )] ∀ h ∈ { 1 , 2 , . . . , H} , ∀ ω ∈ �, ∀ i t ∈ V

ω \ { r ω , s ω } (12)

24 G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35

∑

a ∈ A ω T (k )

f h a = x h k ∀ h ∈ { 1 , 2 , . . . , H} , ∀ ω ∈ �, ∀ k ∈ K

ω (13)

f h a ≤ f h a ′ ∀ h ∈ { 1 , 2 , . . . , H} , ∀ ω ∈ �, ∀ k ∈ K

ω , ∀ a = (i s k , j e k ) ∈ A

ω T (k ) , ∀ a ′ = ( j t , j t ′ ) ∈ A

ω W

, t ≥ e k , t ′ ≤e k +

⌈b k ρ

⌉(14)

x h k ∈ { 0 , 1 } ∀ h ∈ { 1 , 2 , . . . , H} , ∀ ω ∈ �, ∀ k ∈ K

ω (15)

f h a ∈ { 0 , 1 } ∀ h ∈ { 1 , 2 , . . . , H} , ∀ ω ∈ �, ∀ a ∈ A

ω (16)

Equation (7) ensure that exactly one car is assigned to each accepted trip, while constraints (8) make sure that the

assigned car must be purchased (in the first stage). Capacity constraints (9) guarantee that the number of cars that are

simultaneously parked at station i may not exceed its number of charging slots. Observe that final collection arcs need to

be considered on the left-hand side to ensure that the capacity constraints are also met at the end of the planning period.

Constraints (10) ensure that cars may only enter built stations. Equation (11) guarantee that each purchased car is initially

allocated to a station. Thus, we assume that a feasible (free) parking spot at a station must exist for every car that is not used

in a particular scenario, but which performs at least one trip in another scenario. Flow conservation constraints (12) ensure

that the route of each car must correspond to a path through the time-expanded location graph for each scenario. In order

to be feasible, each such path must contain precisely one trip arc from A

ω T (k ) for each trip k ∈ K

ω performed by car h . This

relation, as well as the fact that trip arcs cannot be used by other cars, is guaranteed by equations (11) . Finally, it remains to

ensure that a car may not be used for a trip before its battery is fully recharged. Therefore, forcing constraints (14) guarantee

that a car remains parked (i.e., that there is corresponding flow on the appropriate waiting arcs) for the implied time period

after each performed trip.

5. Heuristic algorithm

The relatively large number of variables and constraints of the ILP formulation introduced in the previous section may

prohibit its successful application to solving medium or large-scale problems as they arise in practice, due to the resulting

long running times or quite high memory requirements. In this section, we therefore propose a heuristic algorithm that can

either be used to quickly obtain solutions for such instances or to provide an initial heuristic solution within an algorithm

solving them with the ILP.

The heuristic, which is detailed in Algorithm 1 , is based on identifying a set of feasible routes for individual cars and

later iteratively including them in the solution, while trying to maximize the resulting expected profit contribution. The

initial set of candidate routes is identified by using a resource-constrained shortest path (RCSP) algorithm in the location

graphs introduced above. Note that we use a variant of an RCSP algorithm that maximizes the obtained profit (sum of

negative arc costs) instead of minimizing the sum of arc costs. Thus, we solve a constrained longest path problem on an

acyclic graph, which is possible in polynomial time, by appropriately changing the dominance rules in the original RCSP

algorithm; more details will be given below. We first observe that a path from r ω to s ω in location graph G

ω represents a

feasible route from the perspective of a single car, if the necessary recharging breaks after each trip are respected. Such a

path can, however, only be added to a current candidate solution if its inclusion (which may involve building new stations

or purchasing an additional car) does not violate the budget constraint or any stations’ capacity constraint.

Two obvious possibilities for iteratively creating a heuristic solution by (iteratively) adding such paths corresponding to

car routes exist:

(i) Find a most profitable, feasible car route by solving the RCSP problem for each scenario (i.e. in each location graph),

add this path, update the location graphs accordingly (considering already performed trips and built stations). Repeat

this process as long as at least one new route is added.

(ii) Identify a set of profitable, feasible car routes by considering all (or a subset of) non-dominated paths, which are

obtained from solving the RCSP problem for the different scenarios, and then try to iteratively add these paths while

respecting the capacity and budget constraints.

We observe that algorithms based on the first option have the advantage of fully considering a partially constructed

solution in each iteration. To this end, information about already open stations, covered trips, and residual capacities of

open stations (at different times of the planning period) may be considered when identifying the next car route by solving

the RCSP problem. On the other hand, it requires the solution of | �| RCSP problems in each iteration (for each car). Despite

the fact that each time-expanded location graph is acyclic and the RCSP is therefore relatively easy to solve by using a label-

setting algorithm (each node needs to be considered only once when expanding its labels for all outgoing arcs), this variant

may yield relatively long running times for large-scale instances.

G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35 25

1 S ′ = ∅ , H

′ = 0 , W

′ = 0 // initialize solution

2 H

′ ω = 0 , K

′ ω = ∅ , ∀ ω ∈ � // used cars, accepted trips per scenario

3 C ω it

= C i , ∀ ω ∈ �, ∀ i ∈ S, ∀ t ∈ { 0 , 1 , . . . , T max } // residual cap. per scen., station, and time

4 for ω ∈ � do // FIRST PHASE

5 ω = ∅ // set of non-dominated paths per scenario

6 ( ̃ V ω , ˜ A

ω ) = (V ω , A

ω )

7 repeat

8 ′ = RCSP (( ̃ V ω , ˜ A

ω )) // find set of paths ⋃

P∈ ′ P = (V ′ , A

′ ) 9 for P = (V ′ , A

′ ) ∈ ′ do

10 ˜ A

ω =

˜ A

ω \ { a ∈

˜ A

ω T (k ) | ∃ k ∈ K

ω s.t. ˜ A

ω T (k ) ∩ A

′ � = ∅} 11 ω = ω ∪ ′ 12 until ′ = ∅ or ˜ A

ω T (k ) = ∅

13 while ⋃

ω∈ � ω � = ∅ do // SECOND PHASE

14 v opt = 0

15 ( ̂ P , ˆ ω , ̂ S , ˆ W , ˆ K ) = ((∅ , ∅ ) , 0 , ∅ , 0 , ∅ ) 16 for ω ∈ � do

17 for P = (V ′ , A

′ ) ∈ ω do

18 valid = true

19 ( ̃ S , ˜ W , ˜ p , ˜ K ) = (∅ , 0 , 0 , ∅ ) // additional stations, budget, profit contr., trips

20 if H

′ ω = H

′ then

˜ W =

˜ W + F car // additional car required

21 for k ∈ K

ω s.t. ∃ (i t , j t ′ ) ∈ A

′ ∩ A

ω T (k ) do

22 if k ∈ K

′ ω then valid = false

23 else

24 ˜ p = ˜ p + p k 25 ˜ K =

˜ K ∪ { k } 26 for (i t , v ) ∈ A

′ \ A

ω I

do

27 if i / ∈ (S ′ ∪

˜ S ) then // additional station(s) required

28 ˜ S =

˜ S ∪ { i } 29 ˜ W =

˜ W + F i 30 else if C ω

it = 0 then valid = false

31 if valid and W

′ +

˜ W ≤ W then

32 if v opt < �ω ˜ p ˜ W + ε then // cmp. efficiency ( ε avoids division through 0)

33 v opt = �ω ˜ p ˜ W + ε

34 ( ̂ P , ˆ ω , ̂ S , ˆ W , ˆ K ) = (P, ω, ̃ S , ˜ W , ˜ K )

35 else ω = ω \ { P } 36 if v opt > 0 then // update solution

37 ˆ ω = ˆ ω \ ˆ P

38 H

′ ˆ ω

= H

′ ˆ ω

+ 1

39 if H

′ < H

′ ˆ ω

then H

′ = H

′ + 1

40 K

′ ˆ ω

= K

′ ˆ ω

∪

ˆ K

41 S ′ = S ′ ∪

ˆ S

42 W

′ = W

′ +

ˆ W

43 for (i t , j t ′ ) ∈ A ( ̂ P ) ∩ (A ̂

ω W

∪ A ̂

ω C ) do // update residual capacities

44 C ̂ ω it

= C ̂ ω it

− 1

Algorithm 1: Path Heuristic.

Algorithm 1 is therefore based on the second option. It solves the RCSP problem for all scenarios ω ∈ � on location

graphs G

ω and stores all non-dominated paths in sets ω in the first phase. Thereby, a standard label-setting algorithm

is used for solving the RCSP problem in each location graph. As noted above, it needs to consider each node i t ∈ V

ω only

once (in non-decreasing order of t ) when extending all labels stored at this node by considering all outgoing arcs from

δ+ (i t ) . Each such label � = ( profit � , break � ) corresponds to a path from r ω to i t and contains information about the profit

contribution profit � of this path and the remaining recharging time break � after which a next trip can be performed. At

each node, dominated labels are removed, i.e., those labels � for which another label � ′ exists at the same node such that

profit � ′ ≥ profit � and break � ′ ≤ break � with at least one of the inequalities being strict. We consider a path P = (V ′ , A

′ ) as non-

dominated if its corresponding label is extended to the sink node without being removed. For each scenario, this process is

repeated on a subgraph

˜ G

ω = ( ̃ V ω , ˜ A

ω ) ⊆ G

ω from which all trip arcs a ∈

˜ A

ω T (k ) corresponding to trips covered by at least

26 G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35

Table 1

Number of created grid instances ( # ) and number of con-

sidered stations (| S |), scenarios (| �|), trips per scenario

(| K ω |) as well as time points in the planning period ( T max )

for each instance size.

# | S | | �| | K ω | T max

small 24 10 ∈ {3, 5} 10 ∈ {15, 30}

medium 24 25 ∈ {3, 5} 50 ∈ {15, 30}

large 24 50 ∈ {3, 5} 100 ∈ {15, 30}

one path from ω have been removed. Non-dominated paths found in the current sub-iteration are temporarily stored in

set ′ and subsequently added to set ω that stores all identified non-dominated paths for scenario ω. The first phase of

the algorithm stops when no further non-dominated paths that contain at least one trip arc could be found and added to

ω .

The second phase of Algorithm 1 considers the set of found paths ⋃

ω∈ � ω in non-increasing order of their efficiency,

computed as the fraction between obtained expected profit contribution and additional costs. In each iteration, these effi-

ciency values are updated first, before adding the most efficient path

ˆ P whose inclusion to the current solution is feasible.

In order to allow for an efficient check as to whether a path can be added to a partial solution, the algorithm keeps track of

purchased cars H

′ (and their usage in different scenarios H

′ ω ), built stations S ′ , remaining budget W − W

′ , residual capacities

of stations C ω it

, and trips K

′ ω covered by already added paths. Thereby, for each considered path P ′ ∈ ω , Algorithm 1 uses ˜ S ,

˜ K , ˜ p and

˜ W to store (additional) needed stations, covered trips, profit contribution and additionally needed budget, respec-

tively, to eventually identify the most efficient path

ˆ P and its corresponding efficiency value v opt in each iteration. Relevant

attributes related to this path

ˆ P are stored in sets ˆ S (stations to build), ˆ K (covered trips) as well as values ˆ W (required bud-

get), and ˆ ω (scenario). Stored attributes are also used to update the residual capacities of every station at each considered

time point. The algorithm terminates after iterating through all previously found paths, yielding built stations S ′ , number of

purchased cars H

′ , the accepted trips per scenario K

′ ω and used budget W

′ .

6. General performance analysis

In this section, we focus on analyzing the performance of the developed algorithms from a general perspective. To allow

insights into the influence of various input parameters, we created a set of instances in which the street network is modeled

as a grid graph and whose parameters are detailed in the following paragraph.

Grid instances. We created benchmark instances in which the street network G = (V, A ) is represented by a grid graph of

dimension 30 × 30. The walking time (in minutes) of each arc a ∈ A is a random integer number between one and five

and the | S | locations of potential stations S ⊆ V are chosen uniformly at random. The capacity C i of each station i ∈ S , is a

randomly chosen integer between one and ten, while the opening costs F i are set to α + βC i where α and β are integers

chosen randomly such that α ∈ [100, 1000] and β ∈ [50, 100], respectively. The following parameter values have been

considered to obtain the final set instances: | �| ∈ {3, 5} with a randomly chosen probability for each scenario, | S | ∈ {10, 25,

50}, | K

ω | ∈ {10, 50, 100} for each scenario ω ∈ �, and T max ∈ {15, 30}. The parameters of each trip k ∈ K

ω are chosen as

follows: origin o k and destination d k are chosen randomly while ensuring that at least one potential station can be reached

with the considered maximum walking time of ten minutes; start time s k randomly from { 0 , . . . , T max − 1 } , end time e k randomly from { s k . . . , T max } , profit contribution p k = 200(e k − s k ) , and a battery consumption of b k = e k − s k while using a

recharging rate of ρ =

10 3 , thus implying a recharging break of 0 . 3(e k − s k ) � after trip k . Six instances have been created

independently for each considered parameter combination. Table 1 summarizes the grid instances created and considered in

our computational study, which are clustered into small, medium , and large instances according to their size. This clustering

is based on our previous experiments ( Kahr, 2016 ) which showed that the performance of the model strongly depends

on the number of trips | K | as well as the number of available cars H but considerably less on the number of stations | S |,

scenarios | �| and the length of the planning period T max .

The following instance-independent parameter values have been used in our experiments. The purchase price of each

car has been set to F car = 100 and four different values have been tested for the budget W that are obtained as fractions

of the overall investment costs necessary to construct all stations and to purchase all available cars, i.e., we used W =w ·

(F car H +

∑

i ∈ S F i )

for w ∈ { 1 10 ,

1 3 ,

1 2 ,

2 3 } , which we refer to as budget fraction . The operating costs of the cars and stations

are chosen relative to their purchasing and constructing costs, respectively, i.e., φ =

F car 100 , ϕ =

F i 70 . The maximum number of

available cars H has been set using the number of cars H

′ used by the heuristic solution (see Section 5 ). Since the developed

heuristic tends to open more stations rather than to buy additional cars, we tested our algorithm with H = 10 H ′ 10 � + H

+ ,for H

+ ∈ { 0 , 10 } , i.e., we rounded up the number of available cars up to the next multiple of ten, and optionally added ten

more cars.

Overall, twelve computational experiments have been performed for each of the 72 grid instances. Each of these 864

experiments has been performed on a single core of a computing cluster consisting of Intel Xeon E5-2670v2 machines with

G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35 27

Table 2

Numbers of solved instances ( # OPT ), average CPU-times ( t avg ) in seconds, and average optimality gaps (gap avg ) in percent

of the exact method with (ILP) and without ( ILP (−) ) initializing it with the heuristic solution, and the heuristic method.

Optimality gaps greater than zero are computed by considering only instances that were not solved optimally. Average

gaps of the heuristic method are relative to the best known solutions. The average CPU-times of the heuristic are only

reported in the case of H + = 0 since we did not run the heuristic again in the case of H + = 10 , but only added 10 more

cars as input parameter for CPLEX. The CPU-times of the exact methods do not include those of the heuristic method.

Results are grouped by the size of the instances, numbers of considered scenarios and numbers of available cars (relative

to the number of cars used by the initial heuristic).

| �| H + # ILP ILP (−) Heuristic

# OPT t avg [ s ] gap avg [%] # OPT t avg [ s ] gap avg [%] t avg [ s ] gap avg [%]

small 3 0 48 48 0.6 0.00 48 0.6 0.00 0.01 24.22

10 48 48 1.5 0.00 48 1.6 0.00 – 24.22

5 0 48 48 1.1 0.00 48 1.2 0.00 0.02 18.46

10 48 48 3.4 0.00 48 3.6 0.00 – 18.46

medium 3 0 48 48 474.3 0.00 48 634.3 0.00 0.08 32.07

10 48 47 1968.0 0.23 48 2 323.1 0.00 – 32.87

5 0 48 47 1 634.0 0.06 47 2 579.6 0.11 0.15 36.42

10 48 45 5 445.4 0.11 45 8 915.3 0.21 – 37.29

large 3 0 48 19 68 175.1 1.20 17 69 102.9 1.50 0.28 40.18

10 48 7 81 937.3 2.28 6 81 379.0 2.15 – 45.08

5 0 48 13 73 373.6 10.15 7 78 326.5 5.10 0.55 43.70

10 48 0 86 400.0 15.32 1 85 125.2 15.08 – 48.61

Table 3

Numbers of solved instances ( # OPT ), average CPU-times ( t avg ) in seconds, and average optimality gaps (gap avg ) in percent

of the exact method with (ILP) and without ( ILP (−) ) initializing it with the heuristic solution, and the heuristic method.

Optimality gaps greater than zero are computed by considering only instances that were not solved optimally. Average gaps

of the heuristic method are relative to the best known solutions. The CPU-times of the exact methods do not include those

of the heuristic method. Results are grouped by the size of the instances, numbers of considered scenarios and numbers of

considered time points.

| �| T max # ILP ILP (−) Heuristic

# OPT t avg [ s ] gap avg [%] # OPT t avg [ s ] gap avg [%] t avg [ s ] gap avg [%]

small 3 15 48 48 1.0 0.00 48 1.0 0.00 0.01 16.9

30 48 48 1.1 0.00 48 1.2 0.00 0.02 31.23

5 15 48 48 1.8 0.00 48 2.0 0.00 0.02 11.08

30 48 48 2.7 0.00 48 2.8 0.00 0.03 25.85

medium 3 15 48 48 900.8 0.00 48 1 596.4 0.00 0.07 33.66

30 48 47 1 541.4 0.23 48 1 361.1 0.00 0.10 31.27

5 15 48 45 3 728.0 0.11 47 4 629.8 0.20 0.11 37.66

30 48 47 3 351.3 0.06 45 6 865.1 0.18 0.18 36.04

large 3 15 48 8 79 195.6 2.13 6 81 424.8 2.27 0.24 42.74

30 48 18 70 916.7 1.44 17 69 057.0 1.34 0.33 42.51

5 15 48 5 81 062.7 14.81 4 81 330.5 11.64 0.50 46.66

30 48 8 78 711.8 11.35 4 81 660.6 8.62 0.61 45.35

2.5 GHz, and a memory limit of 12 GB has been set. Furthermore, an absolute time limit depending on the instance graph

of 20 s (small instances), 8 h (medium instances), or 24 h (large instances) has been applied to each individual run. Our

implementation was done in C++ and uses the IBM ILOG CPLEX Optimizer with concert technology in version 12.6.2. Note that

the heuristic solution has been used to initialize CPLEX in our default setting. To analyze the impact of the initial heuristic,

we also consider variant ILP (−) in which we only use the result of the heuristic to limit the number of available cars (see

above) but do not initialize the exact solver with the computed solution (or its value).

Results overview. We first analyze the numbers of solved instances, average CPU-times and optimality gaps for the three

instance classes (small, medium, large). Thereby, we focus on the influence of the number of scenarios, available cars and

considered time points. The obtained results for small, medium, and large instances are given in Tables 2 and 3 , respectively.

We observe that all small and most medium sized instances could be solved optimally within the given time and memory

limits. The required CPU-time, however, drastically increases with increasing instance size. Consequently, many of the large

instances could not be solved to proven optimality. From Tables 2 and 3 , we conclude that the running times, as well as the

remaining optimality gaps, increase with an increasing number of scenarios. We also observe that instances with a larger

number of cars are much more difficult for our algorithm. All these observations are supported by the fact that increasing

these parameter values (number of scenarios, number of cars) lead to much higher numbers of variables and constraints

to be considered in the corresponding ILP instance. Conversely, the impact of increasing the number of considered time

28 G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35

Fig. 4. Gaps of the heuristic solutions relative to the best known solutions for different instance classes and budget fractions.

points (i.e., the value of T max ) on the algorithm’s performance seems inconsistent. To this end, we observe that the average

CPU-times and remaining optimality gaps even decrease when increasing the number of considered time points for medium

sized instances with | �| = 3 , as well as for large instances. We also conclude that even though most large instances could

not be solved, the remaining optimality gaps seem acceptable from a practical perspective (typically less than 2.5%) as long

as the number of scenarios is small. Thus, we believe that the developed approach is suitable for real-world instances with

not too many scenarios. However, as is common in stochastic optimization, it will be crucial to select an appropriate but

sufficiently small set of scenarios when attempting to solve real world instances. In addition, it will be important to use a

good and not unnecessarily large estimation for the maximum number of cars.

From Tables 2 and 3, we also observe that initializing CPLEX with solutions of the heuristic method typically improves

the performance of the exact method. Overall, a larger number of instances is solved to optimality (seven more instances)

and a comparably large decrease of CPU-times can be observed on medium sized instances. While there also exist cases in

which variant ILP (−) performs better than ILP, the latter seems to be preferable on average. For the computations on the

(comparably large) instances based on real-world data, we therefore chose ILP, i.e., the variant in which CPLEX is initialized

with the solutions of the heuristic method. Thus, we also ensure to obtain valid solutions in case no better solutions can be

found within the given memory and CPU-time limits.

Quality of the initial solution. Supplemental to the overall performance analysis, it seems important to analyze the quality

of the solution computed by the heuristic algorithm introduced in Section 5 . The results will provide insights whether a

reasonable solution quality can be expected when solving very large instances with our heuristic that are too large for the

developed ILP.

Fig. 4 illustrates the gaps of the heuristic solutions relative to the best known solution values in percent. We observe

that the gaps increase with increasing instance size but also with decreasing budget. The latter observation can be ex-

plained by the fact that the heuristic algorithm might tend to open more stations instead of purchasing additional cars.

Overall, it seems that the heuristic derives reasonably good initial solutions for the ILP, but the development of additional

improvement operators should be considered when attempting to solve very large instances with a rather limited budget. An

alternative option with a significantly smaller development effort might be to incorporate some randomization components

in Algorithm 1 and then repeatedly apply the resulting randomized heuristic. If the goal is to identify stations maximizing

the expected profit (or number of satisfied customer requests) without tight budget restrictions, the current heuristic might

be a viable option in practice.

Influence of the available budget. As observed in the previous paragraph, the available budget has a strong impact on the

solution quality achieved by the heuristic algorithm. Thus, we also study its influence on the performance of the overall

algorithms composed of the successive application of the heuristic and the ILP. Fig. 5 contains performance plots showing

the fraction of instances solved within a certain time, as well as the fraction of instances with a given maximum optimality

gap for small, medium, and large instances, respectively.

Similar to the heuristic results, we observe a strong dependence of the performance on the available budget. The difficulty

of an instance, however, does not strictly increase with increasing or decreasing budget. Instead, cases with very low or high

budget can be solved efficiently, while intermediate cases seem more difficult. While this trend is quite clear for small and

medium instances, the small number of solved large instances prevents clear observations for the latter (with respect to

G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35 29

Fig. 5. Fraction of problem instances solved (to optimality) within the given CPU-time limit (in seconds) and remaining optimality gaps (in percent), with

respect to the fraction w (i.e., available budget W relative to the overall investment costs). Time limits were set to 20 s for problem small instances, 8 h

for medium problem instances and to 24 h for large problem instances, respectively.

30 G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35

Table 4

Results of the exact method on instances based on the inner districts of Vienna,

grouped by the number of available cars (relative to the number of cars used

by the initial heuristic) and the available budget as fraction w of the overall

investment costs. Besides the numbers of open stations (| S ′ |) and purchased cars

( H ′ ), we also report the average number of accepted trips per scenario ( ̄K ′ ) as

well as the CPU-time ( t ) in seconds and optimality gaps (gap) in percent of the

exact method (ILP) and the heuristic method. The gaps of the heuristic method

refer to the ratio of the objective values obtained from the heuristic solutions

relative to the best known objective values.

H + w ILP Heuristic

| S ′ | H ′ K̄ ′ t [ s ] gap[%] t [ s ] gap

0 0.10 30 10 76 109 729.0 0.00 25.2 61.84

0.33 49 10 80 4 571.9 0.00 25.2 67.85

0.50 48 10 80 4 486.5 0.00 25.3 61.84

0.66 48 10 80 4 470.9 0.00 25.2 67.86

10 0.10 26 20 94 1 209 600.0 4.91 25.2 61.84

0.33 70 20 107 57 908.0 0.00 25.2 67.86

0.50 70 20 107 51 157.6 0.00 25.3 61.21

0.66 70 20 107 56 762.6 0.00 25.2 64.70

solved instances). With the exception of imposing a very restricted budget (i.e., w = 0 . 1 ) to large instances, we also conclude

that even though only few large instances could be solved optimally, the remaining gaps seem acceptable (below 5%) for at

least 80% of the considered test cases.

7. Case study: Vienna

Besides testing our approach on grid instances, we also applied it to real-world instances based on the city of Vienna

provided to us by the Austrian Institute of Technology (AIT). In this dataset, the street network has been modeled based on

OpenStreetMap data ( OpenStreetMap contributors, 2015 ), which results in a graph representing the whole city and contain-

ing 78 803 vertices and 198 642 arcs. A total of 693 potential stations are located at points of interest, e.g., supermarkets,

parking places, and subway stations. Taxi trips (of a particular week in spring 2014) have been used as an estimation of the

car-sharing demand, since a taxi trip might be substituted by using a car-sharing service in case the latter is easily avail-

able. Origins, destinations, start and end times, as well as estimated battery consumptions (when performing the trip with

an electric vehicle) are associated to each of the 6 640 trips included in the data set 1 . Moreover, the profit contribution of

each trip has been set to 0.3 € per minute. Note that this fee is oriented at the current prices of the car-sharing operator

car2go in Vienna. The construction costs of the stations are based on previous work of the AIT, which has investigated the

introduction of electric taxis in Vienna. Their results indicated that, depending on the concrete location, building a station

costs between 9 0 0 0–64 0 0 0 € as base-price, plus the cost of a slow charging point (17 0 0 0–26 0 0 0 €) times the station’s

capacity. The purchasing costs of a car where estimated at 15 0 0 0 € including quantity discounts.

Seven scenarios were defined (one for each weekday), with probabilities of 0.15 for workdays, 0.13 for Saturday and 0.12

for Sunday, respectively. Note that the planning horizon T max therefore refers to exactly one day and a granularity of one

hour (30 min, 15 min) was used, thus 24 (48, 96) time points are considered for each day and original start and end times

have been rounded down (start time) and up (end time) appropriately. A maximum walking time of 5 min has been used

in our experiments. All tests were performed on the same hardware detailed in the previous section, with a memory limit

of 28 GB and a CPU-time limit of one week.

Besides applying the heuristic to this instance, we performed additional experiments with the exact approach using a

subinstance composed of the eight central districts of Vienna. As a densely populated area with scarce parking spaces and

a relatively large number of inhabitants that do not own a private car, these districts seem to be an ideal region for testing

an electric car-sharing system. The resulting instance contains 13 311 vertices, 32 184 arcs, 201 potential stations and 1 060

trips. It turns out that the exact approach is able to solve this reduced, but still practically relevant instance. Fig. 6 shows

an optimal solution for a particular case in which the available budget was set to 5% of the overall investment costs (i.e.,

w = 0 . 05 ). Tables 4 and 5 summarize the obtained numerical results for the inner districts and whole Vienna, respectively.

From Table 4 , we observe that all available cars are typically used in the exact solution. This confirms the previous

observation that the heuristic tends to buy relatively few cars. All but one of the considered test cases could be solved

to proven optimality. Surprisingly, the number of built stations is usually larger than the number of purchased cars. From

the average numbers of accepted trips, we further conclude that each car typically performs several trips in each scenario.

Furthermore, we conclude that using at most one third of the maximum investment costs (i.e., w = 0 . 33 ) seems sufficient

1 The dataset actually contains 37 965 trips, from which we took the ones for which at least one potential station exists within the chosen maximum

walking distance. Furthermore we excluded trips that do not start and end at the same day, respectively.

G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35 31

Fig. 6. Results of a benchmark instance based on real world data from the inner districts of Vienna. Open stations i ∈ S ′ and the walking distance of 5 min

that users are expected to be willing to walk to/from a station are indicated by the large icons and circles, respectively, whereas the closed stations i ∈ S �S ′ are represented by the small icons. Note that the circles representing the areas covered by open stations are only graphical approximations and the

exact set of nodes reachable within 5 min has been used in the computation.

to obtain a most profitable solution, since the numbers of built stations, purchased cars, and performed trips do not change

for larger values of w .

From Table 5 , we conclude that the solutions produced by the heuristic do not seem to change a lot with increasing

granularity of the planning period. As opposed to the exact results for the inner districts, providing more budget than 33% of

the total investment costs still yields a significant increase in the numbers of opened stations, purchased cars and accepted

trips. This observation may stem from different instance characteristics when also considering the outer parts of Vienna or

from the fact that the heuristic does not make best use of the given budget. Overall, we also observe that a relatively large

portion of the trips are accepted in the heuristic solution if enough budget is available and that the average trip-to-car ratio

per scenario typically is between one and two.

We also provide summarized economic results of the instances based on the inner districts of Vienna given in Table 6 ,

which may act as an exemplary basis for decision-makers. Thereby, we assume an average asset depreciation range of

eight years and thus a depreciation rate of 12.5% per year, as well as a residual value of the capital goods of 5% of the

32 G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35

Table 5

Heuristic results of instances based on the data of

whole Vienna, grouped by the granularity of the

planning period ( T max ) and the available budget as

fraction w of the overall investment costs. Numbers

of open stations (| S ′ |), purchased cars ( H ′ ) and the

average number of accepted trips per scenario ( ̄K ′ ) as well as the CPU-time ( t ) in seconds for solving

the heuristic are given.

T max w | S ′ | H ′ K̄ ′ t [ s ]

24 0.10 88 40 69 9 076.2

0.33 258 173 320 9 087.0

0.50 364 273 459 9 101.8

0.66 461 300 513 9 088.8

48 0.10 81 34 52 11 448.3

0.33 248 163 301 11 454.8

0.50 353 236 429 11 469.6

0.66 450 299 478 11 497.4

96 0.10 88 36 59 10 725.6

0.33 245 156 303 10 734.4

0.50 359 229 428 10 750.7

0.66 454 264 465 10 761.9

Table 6

Summarized economic results of the exact method (ILP) and the

heuristic method on instances based on the inner districts of Vienna.

Besides the average investment costs and yearly operating costs, we

report the average expected profit per year, payoff time and rate of

return per year.

ILP Heuristic

Avg. investment costs 5 560 125 € 1 909 0 0 0 €Avg. operating costs, p.a. 67 648 € 23 226 €Avg. expected profit, p.a. 136 683 € 48 373 €Avg. expected payoff time 6.2 years 6.3 years

Avg. expected rate of return, p.a. 5.5% 4.8%

investment costs, for the computations. Moreover, we assume that an operator can fully take advantage from the depreci-

ation by reducing her/his earnings before taxes from other investment projects. Notice that the results imply that a stand-

alone investment in an electric car-sharing system would be unprofitable in the investigated case without the latter assump-

tion. We further remark that we used a static approach to obtain the reported results and that they also strongly depend on

the model assumptions (given in Section 3 ) in practice. However, the results indicate that the implementation of an electric

car-sharing system in the inner districts of Vienna can be profitable with respect to the underlying assumptions, since the

average expected payoff times are significantly lower than the assumed average asset depreciation range. This may offer op-

portunities for established companies to act as first mover and to gradually expand the operational area, while considering

the expected decreasing costs of electric mobility in the years to come. Furthermore, one can expect the results to improve

if more available cars are considered in the computations, as signified in the previous paragraph. Moreover, we observe

similar results of the exact method and the heuristic method with respect to the relative measurements of profitability (i.e.,

average expected payoff time and rate of return), and therefore conclude that the heuristic method delivers useful results

from an economic perspective, despite the relatively large (optimality) gaps.

8. Conclusions and outlook

In this article, we introduced and studied a stochastic optimization problem that aims to solve the strategic optimization

problem of determining optimal locations for charging stations of (ad-hoc) electric car-sharing systems. We observed that

though such systems may help to overcome important (environmental) challenges arising in cities, there is little scientific

research on the use of electric cars withing car-sharing systems. After stating a couple of practically relevant and important

assumptions (such as availability of an appropriate demand forecast), we gave a formal definition of the resulting two-stage

stochastic optimization problem.

The problem was modeled as an integer linear program in which a set of time-expanded location graphs is used to track

each car’s position during the planning period. We also proposed a heuristic algorithm that is based on the idea of solution

construction by iteratively adding profitable paths that correspond to routes of individual cars. The heuristic has been used

as a stand-alone algorithm for very large scale instances, as well as to provide an initial feasible solution for the subsequent

application of the time-dependent ILP model.

G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35 33

A computational study on a set of grid-graph-based test instances was performed to analyze the influence of different

parameters on the overall performance. The obtained results show that the difficulty of instances increases noticeably with

an increasing number of available cars. The used granularity of the planning period, however, does not seem to have a large

impact on the required solution time. We also showed that instances with a tightly constrained or almost unconstrained

budget are often relatively easy to solve, while those in between (i.e., where the budget is still rather constraining but

already allows to build a significant number of stations and to purchase several cars) seem more difficult.

Finally, we performed a case study on real-world instances from Vienna. It turns out that the developed exact ap-

proach is suitable for solving instances obtained for eight central districts, but cannot be applied to instances in which

the street network models the whole city of Vienna, where we successfully applied the heuristic algorithm. The reason-

ably chosen locations for charging stations on these real-world instances confirm the suitability of the proposed opti-

mization problem. Moreover, we considered an economic perspective and showed that the implementation of an elec-

tric car-sharing system in the inner districts does not seem profitable as a stand-alone investment, but can be prof-

itable next to other projects, which might offer opportunities for established companies to act as first mover in that

business area.

Several possibilities for future research can be derived from our results. From a computational perspective, it might

be worth to develop sophisticated decomposition methods that are likely to yield an exact algorithm with a signifi-

cantly better performance as the current one. Such an algorithm might be suitable for large-scale real-world instances

such as those obtained from considering the whole city of Vienna. Alternatively, one might consider the development

of metaheuristic approaches that might be able to derive better solutions than the greedy heuristic proposed in this ar-

ticle. Besides algorithmic improvements, relevant research directions include the development of models that also con-

sider the relocation of cars by the operator, or that relax the assumption that cars need to be fully recharged before

every trip.

Acknowledgements

The authors thank their project partners from the Austrian Institute of Technology (AIT) for creating the real-world in-

stance from Vienna. This work is supported by the Joint Programme Initiative Urban Europe under the grant 847350 and by

the Austrian Science Fund (FWF) under grant I892-N23 . These supports are greatly acknowledged.

The authors would also like to thank the three anonymous reviewers for their helpful and constructive comments that

helped improve the quality and clarity of the article.

Appendix A. List of sets, variables and parameters

Table 7

Tabular description of the notation used throughout the paper.

Group Notation Description

Input parameters A Arcs of input graph (street network)

B max Battery capacity of each car

b k Battery consumption (overestimated) of trip k ∈ K d k Destination d k ∈ V of trip k ∈ K C i Capacity of a station i ∈ S (maximum number of charging slots)

�k Duration �k = e k − s k of trip k ∈ K e k End time e k ∈ T of trip k ∈ K F i Construction costs of potential station i ∈ S F car Purchasing costs per car

H Number of available cars

S Set of potential stations

ϕ i Operating costs of station i ∈ S φ Operating costs per purchased car

G Input graph (street network) G = (V, A )

K Set of potential trips defined as the union of trips per scenario, i.e., K =

⋃

ω∈ � K ω

each trip k ∈ K is given as tuple ( o k , d k , s k , e k , b k , p k ) representing origin o k , destination d k ,

start time s k , end time e k , battery consumption b k , and profit contribution p k K ω Set of potential trips in scenario ω ∈ �N i Neighborhood of station i ∈ S (set of trip origins and destinations that may be covered by i )

N ( v ) Neighborhood of vertex v ∈ V (set of potential stations within walking distance from v )

o k Origin o k ∈ V of trip k ∈ K

p k Profit contribution of trip k ∈ K

�ω Probability of scenario ω ∈ �( continued on next page )

34 G. Brandstätter et al. / Transportation Research Part B 104 (2017) 17–35

Table 7 ( continued )

Group Notation Description

ρ Rate of recharge (per time unit)

s k Start time s k ∈ T of trip k ∈ K T Planning period T = { 0 , . . . , T max } T max End of planning period

V Nodes of input graph (street network)

W Available budget

� Set of scenarios with probabilities �ω for each ω ∈ �Solution end ( k ) End station of accepted trip k ∈ K ′

H ′ Purchased cars

K ′ Accepted trips

K ′ h (ω) Sequence of trips (in temporal order) performed with car h in scenario ω ∈ �

S ′ Built stations

start ( k ) Start station of accepted trip k ∈ K ′ W

′ Used budget

Variables f h a Whether or not car h ∈ { 1 , . . . , H} travels along arc a ∈ A ω in scenario ω ∈ � (second stage)

x k Whether or not trip k ∈ K is accepted (second stage)

x h k

Whether or not trip k ∈ K is assigned to car h ∈ { 1 , . . . , H} (second stage)

y i Whether or not station i ∈ S is built (first stage)

z h Whether or not car h ∈ { 1 , . . . , H} is purchased (first stage)

Time-exp. graphs A ω Arcs of time-expanded location graph of scenario ω ∈ �A ω I Set of initial allocation arcs of time-expanded location graph in scenario ω ∈ �A ω W

Set of waiting arcs of time-expanded location graph in scenario ω ∈ �A ω C Set of final collection arcs of time-expanded location graph in scenario ω ∈ �A ω T Set of travel arcs of time-expanded location graph in scenario ω ∈ �r ω Artificial root node of time-expanded location graph of scenario ω ∈ �s ω Artificial sink node of time-expanded location graph of scenario ω ∈ �G ω Time-expanded location graph G ω = (V ω , A ω ) of scenario ω ∈ �V ω Nodes of time-expanded location graph of scenario ω ∈ �

Heuristic ˜ A ω Arc set of subgraph ˜ G ω

C ω it

Residual capacity station i ∈ S at time t ∈ T in scenario ω ∈ �˜ G ω Subgraph ˜ G ω = ( ̃ V ω , ̃ A ω ) of time-expanded location graph G ω , ω ∈ �,

obtained by removing all trip arcs corresponding to already covered trips

H ′ ω Number of used cars in scenario ω ∈ � in current (partial) solution ˜ K Set of trips in current path ˆ K Set of trips in currently most efficient path

ω Set of non-dominated paths in scenario ω ∈ �′ Temporary set of non-dominated paths

P Currently considered path ˆ P Most efficient path in current iteration

˜ p Total profit contribution of current path ˜ S Set of additional stations required for current path ˆ S Set of additional stations required for most efficient path ˜ V ω Node set of subgraph ˜ G ω

v opt Efficiency value of currently most efficient path ˜ W Required budget for current path ˆ W Required budget for currently most efficient path

General δ+ (i t ) Set of outgoing arcs of vertex i t ∈ V ω in time-expanded location graph G ω , ω ∈ �

δ−(i t ) Set of ingoing arcs of vertex i t ∈ V ω in time-expanded location graph G ω , ω ∈ �

f h [ A ′ ] Sum of flow variables over all arcs in subset A ′ ⊂ A ω , i.e.,∑

a ∈ A ′ f h a

H + Additional cars added to the number of cars in the heuristic solution

(rounded up to the next multiple of ten)

w Budget fraction (available budget relative to overall investment costs)

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