Determining optimal locations for charging stations of electriccar-sharing systems under stochastic demand

Georg Brandstätter, Michael Kahr and Markus Leitner

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

georg.brandstatter@univie.ac.at, m.kahr@univie.ac.at,markus.leitner@univie.ac.at

March 15, 2017(first version: November 6, 2016)

AbstractIn this article, we introduce and study a two-stage stochastic optimization problem suitable to

solve strategic optimization problems of car-sharing systems that utilize electric cars. By combiningthe individual advantages of car-sharing and electric vehicles, such electric car-sharing systems mayhelp 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 optimizationproblem are developed and tested on real world instances from the city of Vienna, as well as on grid-graph-based instances. An analysis of the influence of different parameters on the overall performanceand managerial insights are given. Results show that the developed exact approach is suitable formedium sized instances such as the ones obtained from the inner districts of Vienna. They alsoshow that the heuristic can be used to tackle very-large-scale instances that cannot be approachedsuccessfully by the integer-programming-based method.

Keywords: location analysis, car-sharing, electric cars, time-dependent formulations, integerlinear programming, stochastic optimization

1 IntroductionThe expected growth of the human population by two to four billion in the first half of the 21st centurywill impose severe challenges to humanity. Some of them are intensified by the increasing trend towardsurbanization, especially in developing countries [14]. Two main sources of such challenges are the expectedlarge increase in demand of energy and transportation. Without significant changes in the mode oftransportation and type of fuels used, increased demands may lead to shortages of fossil fuels, whichare still the dominating sources of energy and are estimated to be exhausted before 2050 [37]. Inaddition, severely amplified problems with respect to pollution, congestion, noise and lack of parkingspace are expected. These challenges might be partly met by the consideration and implementation ofnew concepts of transportation such as car-sharing systems, which can reduce the number of circulatingcars [29], as well as the total distance traveled by them, see Shaheen et al. [38]. Hence, such systems arelikely to reduce congestion-related delays and to free up parking space [15]. A possibility to overcomethe rapid exhaustion of fossil fuels and therefore to decrease the emissions of greenhouse gases is theincreased usage of electric-powered vehicles (EVs), premised that the electric power comes from cleanenergy sources [22]. The fact that the market-share of EVs was extremely low (0.1%) compared to thenumber of all passenger vehicles worldwide in 2015 underlines their potential [1]. However, a majordisadvantage of EVs is the large amount of time needed for recharging them compared to the amount oftime needed to refuel conventional vehicles, and the lack of (private) charging stations in urban areas,which impedes the growth of privately held EVs. A successful implementation of car-sharing systems

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that utilize EVs, i.e., electric car-sharing systems, may help overcome the aforementioned environmentalchallenges as they combine the advantages of car-sharing and electric vehicles.

Operators or cities supporting electric car-sharing systems must, however, address difficult strategicquestions before possibly opening their business. Besides selecting an appropriate mode and area ofoperation, these particularly include the question of where to build charging stations at which carscan be recharged while not used by customers. Additional aspects that shall be considered includethe number of purchased cars and their (initial) distribution over the operational area in order to bestmeet customer demands. To this end, several car-sharing concepts are known, which we summarize inthe following paragraph where we also address the question of their applicability to electric car-sharingsystems.

Car-sharing concepts One main classification of car-sharing systems describes whether users are onlyallowed to pick up and drop off cars at designated stations (i.e., station-based systems) or whether tripscan, in principle, be started and ended at any free parking spot in the operational area (i.e., free-floatingsystems). Both variants can be implemented either as a traditional system in which pre-bookings forpredefined time periods are required, which include the specification of pick-up and drop-off locations, oras ad-hoc systems that do not require pre-bookings. Note that ad-hoc systems may nevertheless includethe possibility of reserving a vehicle for a short period of time before starting a trip, thus avoiding thecase where an available car is taken by another customer in the meantime. Finally, while most systemsinclude the possibility of one-way trips, others are restricted to round-trips for which the pick-up anddrop-off location needs to coincide, see, e.g., Boyaci et al. [6].

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 foroperators in the context of electric car-sharing. Each station can be equipped with charging slots atwhich idle cars can be recharged. Another benefit is that the pre-booking process can help prevent bothshortages 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 vehicleis 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 givenamount of time, and reserve a free parking slot at the desired destination). Along the same lines, severaldisadvantages of free-floating electric car-sharing systems are observed. Frequent recharging of emptyvehicles and the poor predictability of shortages and excess supply in specific areas require the use ofpowerful user- or operator-based relocation strategies, see, e.g. Barth et al. [4], Kek et al. [24].

Contribution and outline In this article, we introduce a combinatorial optimization problem tar-geting the strategic planning process of electric car-sharing systems in order to appropriately supportdecision makers. Given a stochastic demand forecast, its aim is to identify optimal locations for chargingstations and an associated number of required EVs in order maximize the expected profit, obtained fromaccepting trips in a predefined planning period. After a brief literature review in Section 2, the newproblem is described in detail and formally defined in Section 3. A two-stage stochastic integer linearprogram (ILP) for the considered problem in its deterministic equivalent is introduced in Section 4, whileSection 5 details a heuristic solution approach. The latter is used to either compute an initial solutionwhen solving a benchmark instance to optimality with our ILP formulation, or as a standalone heuristicto compute good, but typically not optimal solutions to large-scale instances. Computational resultsobtained from solving grid-graph instances are discussed in Section 6, where we focus on a more generalperformance analysis and the influence of different input parameters. In Section 7, we discuss results andfindings obtained from applying the developed methods to real world instances from Vienna and providemanagerial insights. Conclusions are drawn in Section 8, where we also point out directions for futureresearch. Finally, Appendix A contains a list of the most important sets, variables and parameters usedthroughout this article.

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2 Previous and related workThough an increased interest in electric car-sharing systems can be observed recently, the existing scien-tific literature related to optimization problems in such systems is still relatively scarce. Several articlesdo, however, address the case of privately owned or commercial electric vehicles. A more in-depthoverview of the current state of the art regarding optimization problems arising in the context of electriccar-sharing, as well as the aforementioned related areas, is given by Brandstätter et al. [7] in their recentsurvey on the topic.

We first summarize relevant literature related to the placement of public charging stations for privatelyowned electric vehicles, before turning our attention to literature on (electric) car-sharing. Worley et al.[39] developed an integer linear program that determines optimal locations for charging stations for fleetowners and simultaneously suggested routes for the vehicles. Instead of maximizing profits, the givenmodel minimizes the total costs. A mixed integer linear program that minimizes the total access coststo charging stations, based on walking distances was introduced by Chen et al. [13]. They used datafrom over 30 000 records of personal trips. Nie and Ghamami [32] investigated how to select the batterysize and the capacity in terms of the number of charging stations and charging power needed, in orderto meet a given level of service. Their objective was to minimize the social costs. A genetic algorithmto find (sub)optimal locations for public charging stations for EVs was developed by Dong et al. [17],who also provided a case study based on multiday GPS-based travel survey data. They showed thatthe traveled miles and the number of trips of EVs could be significantly increased by installing publiccharging stations at popular destinations, with reasonable infrastructure investment. Another relatedstudy has been recently performed by Faridimehr et al. [21]. They proposed a two-stage stochastic ILP(tackled via sample-average approximation) and a heuristic algorithm with the goal of maximizing thedemand of private vehicles (trips) that can be covered through the established charging stations. Datauncertainty stemming from various sources (e.g., arrival and departure times) is considered. Resultsfrom a case study in Detroit midtown (Michigan, US) are reported.

Barth and Todd [3] proposed a queuing-based discrete-event simulation model for electric car-sharingsystems with the objective to analyze operational issues. They state that the most effective numberof vehicles is in the range of 3-6 per 100 trips based on a 24h day. If the objective is to minimize thenumber of relocations, they suggest that the number of vehicles should be approximately 18-24 per 100trips. Their model shows that the steadiness of the operation of car-sharing systems is most sensitive tothe car-to-trip ratio. Cepolina and Farina [12] studied a multi-station electric car-sharing system usingreal-word data from Genoa. They provided an optimization model to determine the dimension of thefleet and the distribution over the stations with the objective of minimizing the overall costs, i.e., costs ofthe transportation system and the users costs, whereby the latter depend on waiting times. Rechargingof the vehicles is assumed to happen at idle times. An existing one-way, pre-booked electric car-sharingsystem in Kyoto was investigated by Nakayama et al. [31]. They developed a simulation model with theobjective of maximizing the check-outs of the EVs, with the number of vehicles, capacities of the stationsand the number of users as decision variables. Results suggest that the optimal number of vehicles isabout half of the total amount of parking slots. Boyaci et al. [6] presented a multi-objective integerlinear 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 ourapproach enforces necessary recharging stops based on the actually fulfilled demand, theirs relies on itbeing known a priori. Li et al. [26] approached the problem of minimizing the overall system costs ofone-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 thatdecomposes the studied area into a number of small neighborhoods such that each can be approximatedby an infinite homogeneous hyperplane. They showed that the solutions of this method are able toapproximate those of its discrete counterpart efficiently and with high accuracy, even for large-scaleheterogeneous problems. Moreover, they performed a case study using the transportation network ofSioux-Falls city (North Dakota, US) and drew several managerial insights. Recently, Brandstätter et al.[8] introduced and studied an optimization problem targeting the planning of charging stations of electriccar-sharing systems. The authors present several ILP formulations and two heuristic methods, developappropriate solution algorithms, and compare them empirically with respect to their performance. Incontrast to the present article, they consider the case of deterministic demands and also focus on the

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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 inves-

tigated by Rickenberg et al. [35]. They suggested a mixed integer linear program with the objective ofminimizing the total costs while satisfying customer demand and preferences. Furthermore they assumeda stochastic demand modeled by a normal distribution.

In conclusion, we observe that most research including stochastic demand in the context of car-sharingfocuses on vehicle relocation in order to prevent shortages and excess supply in parts of the service area,see, e.g., [4, 9, 24]. 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 stochasticdemand, which happens at the strategic planning level.

A problem related to optimal placement of charging stations in urban areas deals with the placementof such stations between cities. In that context, trips undertaken by customers are usually too long tobe feasible on a single battery charge, which necessitates en-route recharging. Models and algorithmsfor solving these intercity charging station location problems are described by Arslan and Karaşan [2],as well as by Kuby and Lim [25] and Capar et al. [10] who consider the analogous problem of placingrefueling stations for alternative-fuel vehicles. In these articles, the authors seek to cover the rechargingor refueling demand of trips by placing appropriate stations along them instead of at their start orend, which is opposite to the requirements of an urban car-sharing system like the one considered inthe present article. Furthermore, none of these articles considers the use-case of car-sharing and theassociated modeling of available cars at individual stations.

The problem of finding locations for bike-sharing stations is closely related to that of finding suchlocations for charging stations in a station-based one-way electric car-sharing system. This problem, inconjunction with that of optimizing the network structure of bike paths between the stations and thecorresponding routing of bicycles, is studied by, among others, Lin and Yang [27] and Lin et al. [28].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 mightrun out of battery during trips if one would simply adopt the obtained solutions. Furthermore, theauthors of both articles consider only yearly travel demands between points in their models, which cannotaccount for hourly, daily or seasonal differences in demand. Station capacities are assigned in such a wayas to cover the daily demands (that is assumed to follow a certain probability distribution) with a certainprobability. Even algorithms that incorporate electric bicycles, like the one proposed by Martinez et al.[30], do not consider provisioning for such recharging breaks, which have been an important considerationin other optimization problems dealing with electric vehicles, such as the electric vehicle-routing problemwith time windows and recharging stations proposed by Schneider et al. [36]. The authors also assumethat all demands travel along the shortest paths from origins to destinations, the lengths of which alsodetermine duration and profit of each trip. Such a demand model does not allow for the incorporation ofround-trips or trips with multiple stops or detours (which might, for instance, occur when customers usecar-sharing for shopping trips), or for the implementation of more flexible pricing schemes with discountsfor longer rentals (which are used by major car-sharing providers). None of the aforementioned articles onbike-sharing station placement perform a detailed computational study where the presented algorithm’sperformance and its dependence on various instance characteristics is analyzed. Another frequentlyconsidered optimization problem related to bicycle sharing is concerned with re-balancing the system byrelocating bicycles between stations. Both exact and heuristic algorithms for solving this problem havebeen proposed, see, e.g., Erdoğan et al. [19, 20] or Di Gaspero et al. [16]. In these articles, the set ofcharging 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 asby a truck transporting them), it remains unclear whether these algorithms can be easily adapted to theproblem of re-balancing electric car-sharing systems where this is not the case.

Finally, we note that the problem studied in this article is closely related to a special case of theone-to-one vehicle routing problems with pickup and delivery and stochastic demand, also known asdial-a-ride problem in the literature, see, e.g., [5, 34] for relatively recent surveys. In this special casewe restrict time windows for pick-up and delivery to discrete time points, consider only one commodity,set the vehicle capacities to one (i.e., each cargo has to be delivered immediately after its pick-up), andconsider service times at the destination of each commodity equal to the required recharging breaks. In

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contrast to the dial-a-ride problem, we do consider, however, multiple potential start and/or end pointsfor each trip (commodity). In addition, we do not allow vehicles to travel without carrying a commodityas the latter would correspond to a car moving without performing a trip in our setting.

3 Problem definitionAs mentioned in the introduction, the present work focuses on decision support at the strategic planninglevel of electric 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 electriccars to purchase in order to maximize the expected profit during operation, which is earned from acceptedcustomer trips. Thereby, we focus on a one-way, station-based system. Note that our model is able toaddress ad-hoc car-sharing systems as well as those that require pre-bookings. This observation stemsfrom the fact that strategic planning is performed based on estimated customer demands that may, forexample, be obtained from historical data or surveys. Once stations are built, an operator may decideon a more traditional variant with required pre-bookings or an ad-hoc system that will be harder tomanage.

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 asso-ciated probabilities. Each scenario is given as a set of estimated trips that account for spatial andtemporal 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 batteryconsumption and profit contribution.

• The mode of planning is conservative in the sense that each selected trip needs to be assigned to aninitially fully charged vehicle. While this assumption is likely to be relaxed in operation, it ensuresthat small mistakes in the estimation of battery consumptions (see above) will not have severeimpacts in the sense that a car will run out of battery during a trip. To this end, note that in urbanareas trips are usually battery-feasible when started with a fully charged car, see Duchrow et al. [18].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 wouldinduce rather long waiting times that are unacceptable for intra-city trips. For simplicity, we alsoassume linear recharging characteristics (with respect to time). Our approach is, however, flexibleenough to consider arbitrary (non-linear) charging functions as long as the required time to fullyrecharge a vehicle (from some arbitrary, but known, battery state) can be precomputed. Theseassumptions (linear charging and fully recharging a vehicle at each recharging stop) are also in linewith the assumptions made for other optimization problems using electric cars, such as the electricvehicle-routing problem with time windows and recharging stations, see Schneider et al. [36].

• The potential locations for charging stations have individual maximum capacities (i.e., maximumnumbers of charging slots) which are subject to local conditions. If a station is built, it is equippedwith the maximum number of charging slots possible.

• Customers are willing to walk from their origin to a charging station and from a charging stationto their destination, as long as the associated walking distance (time) does not exceed a giventhreshold.

• 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 thestreet network of the potential operational area. Set S ⊆ V describes the potential locations of chargingstations where each i ∈ S has associated construction (building) costs Fi ≥ 0, operating costs ϕi ≥ 0,and a capacity Ci ∈ N describing the number of charging slots that can be built at that station. For eachstation i ∈ S, its neighborhood Ni ⊆ 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

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the neighborhood of a vertex v ∈ V is defined as N(v) = i ∈ S | v ∈ Ni. Parameter H ∈ N defines themaximum number of available cars, each of which has acquisition costs Fcar ≥ 0, operating costs φ ≥ 0,a battery capacity of Bmax, and a recharging rate of ρ, 0 < ρ ≤ Bmax, per time unit. The availablebudget 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, . . . , Tmax ofdiscrete time points in the planning period. Let Ω be the set of demand scenarios and Ψω > 0 bethe probability of scenario ω ∈ Ω such that

∑ω∈Ω Ψω = 1. Each scenario ω ∈ Ω consists of a set of

trips Kω, whereby each trip k ∈ Kω is given as a tuple (ok, dk, sk, ek, pk, bk). Thereby, for each tripk ∈ K =

⋃ω∈ΩK

ω, ok ∈ V and dk ∈ V denote the origin and destination of trip k, while sk ∈ T andek ∈ T are the associated start and end times. Notice that sk < ek and that we denote by ∆k = ek − skthe duration of trip k. Finally, pk > 0 is the profit contribution of trip k representing its revenue reducedby its (estimated) variable costs while bk, 0 ≤ bk ≤ Bmax, is its (estimated) battery consumption. Asmentioned above, we assume that an operator is able to compute an upper bound on a trip’s batteryconsumption based on its duration ∆k, its origin ok, and its destination dk, respectively. We observethat the time required to fully recharge a car after trip k is given as

⌈bk

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 purchased 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′ Fi+FcarH

′ ≤W . For each scenario ω ∈ Ω, a trip k ∈ Kω can only be accepted if it is assigned(a) a purchased car h, 1 ≤ h ≤ H ′, (b) a built start station start(k) ∈ N(ok) ∩ S′ in the neighborhoodof its origin, and (c) a built end station end(k) ∈ N(dk) ∩ S′ in the neighborhood of its destination.Thus, from a users’ perspective, trip k consists of first walking from ok to station start(k), driving theassigned car to station end(k), and finally walking from end(k) to dk, see Figure 1. For each ω ∈ Ω andeach purchased car h, 1 ≤ h ≤ H ′, let K ′h(ω) = (kω1 , kω2 , . . . , kωl ) be the sequence of trips performed withcar h in temporal order, i.e., ekω

j≤ skω

j+1, ∀j ∈ 1, 2, . . . , l− 1. To be feasible, each such sequence must

satisfy the following conditions: (a) the end-station of a trip must be the start station of the subsequenttrip, i.e., end(kωj ) = start(kωj+1), 1 ≤ j < l; (b) all used start- and end-stations must be built, i.e.,⋃lj=1

(start(kωj ), end(kωj )

)⊆ S′; and (c) the temporal break between any two successive trips must be

sufficient to fully recharge the car, i.e., skωj+1≥ ekω

j+ dbkω

j/ρe, 1 ≤ j < l. Finally, each solution must also

meet the capacity constraints imposed by built stations. Thus, for each scenario ω ∈ Ω the number ofcars that are simultaneously at station i ∈ S′ may not exceed its number of charging slots Ci during theplanning period. To this end, car h is at time t at station start(kω1 ) if 0 ≤ t < skω

1, at station start(kωj ),

2 ≤ j ≤ l, if ekωj−1≤ t < skω

j, and at station end(kωl ) if ekω

l≤ t ≤ Tmax. Figure 2 shows an instance

together with a feasible solution and also indicates the trips realized in the two scenarios considered.

i1

i2

i3

k

ok

dk

(a)

i1

i2

i3k

ok

dk

(b)

Figure 1: Two options for routing a trip k from origin ok to destination dk when assuming that stationsi1, i2 and i3 are built (i.e., i1, i2, i3 ⊆ S′) and that stations i2 and i3 are within walking distance ofdk (i.e., i2, i3 ⊆ N(dk)). (a) Client returns the car at station i2; (b) Client returns the car at stationi3. Solid arcs represent the trip taken by the rental car, while dashed arcs indicate walking from / tostations.

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i1

i4

i2

i3

k1 1

k12

k 13

k21

k22

k 23

(a) Instance (user perspective)

i1

i4

i2

k1 1

k 13

k21

k22

k 23

(b) Solution (operator perspective)

Figure 2: (a) Problem instance with two scenarios ω ∈ Ω of equal probability, and (b) optimal solution tothis instance. Each potential station i ∈ S has a construction cost Fi = 1000 and cars can be purchasedat the cost Fcar = 100, but there is only a budget W of 3500 available. The trips k ∈ Kω are givenin temporal order such that ekω

j< skω

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 budgetlimitation, station i3 cannot be constructed, and therefore trip k1

2 must be discarded. Note that onlyone car is purchased in the optimal solution.

4 Time-dependent integer linear programIn this section, we introduce a two-stage stochastic integer linear programming formulation for the SCSLPin its deterministic equivalent form. Formulation (1)–(5) uses the following two sets of first-stage decisionvariables: (i) variables yi ∈ 0, 1, ∀i ∈ S, indicate whether a station is built or not, while (ii) variableszh ∈ 0, 1, ∀h ∈ 1, 2, . . . ,H, indicate whether or not car h is purchased. Finally, variables xk ∈ 0, 1,∀k ∈ Kω, ∀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)–(5) in order to define the objective function (1). Furthersecond-stage decision variables will be introduced below.

max∑ω∈Ω

Ψω

( ∑k∈Kω

pkxk

)−∑i∈S

ϕiyi −∑h∈H

φzh (1)

s.t.∑i∈S

Fiyi +∑h∈H

Fcarzh ≤W (2)

(x,y, z) ∈ Xω ∀ω ∈ Ω (3)xk ∈ 0, 1 ∀k ∈ Kω, ∀ω ∈ Ω (4)yi ∈ 0, 1 ∀i ∈ S (5)zh ∈ 0, 1 ∀h ∈ 1, 2, . . . ,H (6)

The objective function (1) maximizes the expected (second-stage) profit contribution of the acceptedtrips reduced by the operating costs of the built stations and purchased cars. The budget constraint (2)accounts for the limited budgetW . For each scenario ω ∈ Ω, abstract constraints (3) are used to state thatthere must be a way to extend the partial solution implied by the first stage decisions (stations, cars) andthe accepted trips to a feasible solution of the SCSLP. To this end, set Xω 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 tripsuch that the trip sequence associated to each car must be a feasible route. In addition, the capacityconstraints imposed by the built stations must be met by the union of these routes in each scenario, seethe solution description in Section 3 for more details. Our ILP formulation for modeling constraints (3)is based on tracking each cars position at each point in time and in each scenario, by considering thefollowing 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, weintroduce a time-expanded location graph Gω = (V ω, Aω) for each scenario ω ∈ Ω. Node set V ω consists

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of a source (root) node r, a sink node s, and one node it for each station i ∈ S and each consideredtime point t ∈ 0, 1, . . . Tmax. Arc set Aω is the union of waiting arc AωW = (it, it+1) | i ∈ S, t ∈0, 1, . . . , Tmax − 1, travel arcs AωT =

⋃k∈Kω AωT(k), initial allocation arcs AωI = (r, i0) | i ∈ S, and

final collection arcs AωC = (iTmax , s) | i ∈ S. Thereby, AωT(k) = (isk, jek

) | i ∈ N(ok), j ∈ N(dk) isthe set of trip arcs corresponding to trip k ∈ Kω. Note that a time-expanded location graph containsparallel travel arcs if two or more trips in the same scenario have identical start- and end-times andif, additionally, their sets of potential start- and end-stations, respectively, overlap. Waiting arcs willbe 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 ina solution will be interpreted as initially placing cars at the corresponding station, while final collectionarcs 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 willbe linked to each available car. An example time-expanded location graph and a solution correspondingto scenario one of Figure 2 is given in Figure 3.

rω sω

sk1

ek1 sk2

ek2

sk3

ek2

i1 ∈ S′

i2 ∈ S′

i3 ∈ S

i4 ∈ S′

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8 t = Tmax

k 11

k1 3

k12

Figure 3: Time-expanded location graph for the first scenario of the instance given in Figure 2a. Thesecond-stage solution from Figure 2b is indicated by bold arcs. Stations i1, i2, i4 are opened, i.e.,S′ = i1, i2, i4, while potential station i3 remains closed. Consequently, trip k1

2, which is indicated bythe dotted line, cannot be performed.

We define second-stage flow variables fha ∈ 0, 1, ∀h ∈ 1, 2, . . . ,H, ∀a = (it, jt′) ∈ Aω, ∀ω ∈ Ω,to indicate whether car h travels from station i at time t to station j at time t′. Note that travelingalong a waiting arc (it, it′) corresponds to parking (and recharging) a car at station i from time t untiltime t′. Additionally, we also use variables xhk ∈ 0, 1, ∀h ∈ 1, 2, . . . ,H, ∀ω ∈ Ω, ∀k ∈ Kω, thatwill 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 thisformulation, for each scenario ω ∈ Ω and each node u ∈ V ω, we also use notations δ+(u) = (u, v) ∈ Aωand δ−(u) = (v, u) ∈ Aω to refer to the set of outgoing and incoming arcs of a node u, respectively.For a subset of arcs A′ ⊂ Aω, we also use notation fh[A′] =

∑a∈A′ f

ha .

H∑h=1

xhk = xk ∀ω ∈ Ω, k ∈ Kω (7)

xhk ≤ zh ∀h ∈ 1, 2, . . . ,H, ∀ω ∈ Ω, ∀k ∈ Kω (8)H∑h=1

∑a∈δ+(it)∩(Aω

W∪AωC)

fha ≤ Ciyi ∀ω ∈ Ω, ∀it ∈ V ω \ rω, sω (9)

fh[δ−(it)] ≤ yi ∀h ∈ 1, 2, . . . ,H, ∀ω ∈ Ω, ∀it ∈ V ω \ rω, sω (10)fh[AωI ] = zh ∀h ∈ 1, 2, . . . ,H, ∀ω ∈ Ω (11)

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fh[δ−(it)] = fh[δ+(it)] ∀h ∈ 1, 2, . . . ,H,∀ω ∈ Ω, ∀it ∈ V ω \ rω, sω (12)∑a∈Aω

T(k)

fha = xhk ∀h ∈ 1, 2, . . . ,H, ∀ω ∈ Ω, ∀k ∈ Kω (13)

fha ≤ fha′

∀h ∈ 1, 2, . . . ,H, ∀ω ∈ Ω, ∀k ∈ Kω,

∀a = (isk, jek

) ∈ AωT(k), ∀a′ = (jt, jt′) ∈ AωW,

t ≥ ek, t′ ≤ ek +⌈bkρ

⌉ (14)

xhk ∈ 0, 1 ∀h ∈ 1, 2, . . . ,H, ∀ω ∈ Ω, ∀k ∈ Kω (15)fha ∈ 0, 1 ∀h ∈ 1, 2, . . . ,H, ∀ω ∈ Ω, ∀a ∈ Aω (16)

Equations (7) ensure that exactly one car is assigned to each accepted trip, while constraints (8) makesure that the assigned car must be purchased (in the first stage). Capacity constraints (9) guarantee thatthe number of cars that are simultaneously parked at station i may not exceed its number of chargingslots. Observe that final collection arcs need to be considered on the left-hand side to ensure that thecapacity constraints are also met at the end of the planning period. Constraints (10) ensure that carsmay only enter built stations. Equations (11) guarantee that each purchased car is initially allocatedto a station. Thus, we assume that a feasible (free) parking spot at a station must exist for every carthat is not used in a particular scenario, but which performs at least one trip in another scenario. Flowconservation constraints (12) ensure that the route of each car must correspond to a path through thetime-expanded location graph for each scenario. In order to be feasible, each such path must containprecisely one trip arc from AωT(k) for each trip k ∈ Kω performed by car h. This relation, as wellas the fact that trip arcs cannot be used by other cars, is guaranteed by equations (13). Finally, itremains 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 onthe appropriate waiting arcs) for the implied time period after each performed trip.

5 Heuristic algorithmThe relatively large number of variables and constraints of the ILP formulation introduced in the previoussection may prohibit its successful application to solving medium or large-scale problems as they arisein 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 suchinstances or to provide an initial heuristic solution within an algorithm solving them with the time-expanded ILP.

The heuristic, which is detailed in Algorithm 1, is based on identifying a set of feasible routes forindividual cars and later iteratively including them in the solution, while trying to maximize the result-ing 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 usea variant of an RCSP algorithm that maximizes the obtained profit (sum of negative arc costs) insteadof minimizing the sum of arc costs. Thus, we solve a constrained longest path problem on an acyclicgraph which is possible in polynomial time, by appropriately changing the dominance rules in the originalRCSP algorithm; more details will be given below. We first observe that a path from rω to sω in locationgraph Gω represents a feasible route from the perspective of a single car, if the necessary rechargingbreaks after each trip are respected. Such a path can, however, only be added to a current candidatesolution if its inclusion (which may involve building new stations or purchasing an additional car) doesnot violate the budget constraint or any stations’ capacity constraint.

Two obvious possibilities for iteratively creating a heuristic solution by (iteratively) adding such pathscorresponding to car routes exist:

(i) find a most profitable, feasible car route by solving the RCSP problem for each scenario (i.e. ineach location graph), add this path, update the location graphs accordingly (considering alreadyperformed trips and built stations). Repeat this process as long as at least one new route is added.

9

(ii) identify a set of profitable, feasible car routes by considering all (or a subset of) non-dominatedpaths, that are obtained from solving the RCSP problem for the different scenarios and then tryto 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 apartially 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) maybe considered when identifying the next car route by solving the RCSP problem. On the other hand, itrequires the solution of |Ω| RCSP problems in each iteration (for each car). Despite the fact that eachtime-expanded location graph is acyclic and the RCSP is therefore relatively easy to solve by using alabel-setting algorithm (each node needs to be considered only once when expanding its labels for alloutgoing arcs), this variant may yield relatively long running times for large-scale instances.

Algorithm 1 is therefore based on the second option. It solves the RCSP problem for all scenariosω ∈ Ω on location graphs Gω and stores all non-dominated paths in sets Πω in the first phase. Thereby,a standard label-setting algorithm is used for solving the RCSP problem in each location graph. Asnoted above, it needs to consider each node it ∈ V ω only once (in non-decreasing order of t) whenextending all labels stored at this node by considering all outgoing arcs from δ+(it). Each such label` = (profit`, break`) corresponds to a path from rω to it and contains information about the profitcontribution profit` of this path and the remaining recharging time break` after which a next trip canbe 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 theinequalities being strict. We consider a path P = (V ′, A′) as non-dominated if its corresponding labelis extended to the sink node without being removed. For each scenario, this process is repeated on asubgraph Gω = (V ω, Aω) ⊆ Gω from which all trip arcs a ∈ AωT(k) corresponding to trips covered by atleast one path from Πω have been removed. Non-dominated paths found in the current sub-iteration aretemporarily stored in set Π′ and subsequently added to set Πω that stores all identified non-dominatedpaths for scenario ω. The first phase of the algorithm stops, when no further non-dominated paths thatcontain at least one trip arc could be found and added to Πω.

The second phase of Algorithm 1 considers the set of found paths⋃ω∈Ω Πω in non-increasing order of

their efficiency, computed as the fraction between obtained expected profit contribution and additionalcosts. In each iteration, these efficiency values are updated first, before adding the most efficient path Pwhose inclusion to the current solution is feasible. In order to allow for an efficient check as to whethera path can be added to a partial solution, the algorithm keeps track of purchased cars H ′ (and theirusage in different scenarios H ′ω), built stations S′, remaining budget W − W ′, residual capacities ofstations Cωit, and trips K ′ω covered by already added paths. Thereby, for each considered path P ′ ∈ Πω,Algorithm 1 uses S, K, p and W to store (additional) needed stations, covered trips, profit contributionand additionally needed budget, respectively, to eventually identify the most efficient path P and itscorresponding efficiency value vopt in each iteration. Relevant attributes related to this path P arestored in sets S (stations to build), K (covered trips) as well as values W (required budget), and ω(scenario). Stored attributes are also used to update the residual capacities of every station at eachconsidered 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 usedbudget W ′.

6 General performance analysisIn this section, we focus on analyzing the performance of the developed algorithms from a generalperspective. To allow insights into the influence of various input parameters, we created a set of instancesin which the street network is modeled as a grid graph and whose parameters are detailed in the followingparagraph.

Grid instances We created benchmark instances in which the street network G = (V,A) is representedby a grid graph of dimension 30×30. The walking time (in minutes) of each arc a ∈ A is a random integernumber between one and five and the |S| locations of potential stations S ⊆ V are chosen uniformly atrandom. The capacity Ci of each station i ∈ S, is a randomly chosen integer between one and ten,

10

1 S′ = ∅, H ′ = 0, W ′ = 0 // initialize solution2 H ′ω = 0, K ′ω = ∅, ∀ω ∈ Ω // used cars, accepted trips per scenario3 Cωit = Ci, ∀ω ∈ Ω, ∀i ∈ S, ∀t ∈ 0, 1, . . . , Tmax // residual cap. per scen., station, and

time4 for ω ∈ Ω do // FIRST PHASE5 Πω = ∅ // set of non-dominated paths per scenario6 (V ω, Aω) = (V ω, Aω)7 repeat8 Π′ = RCSP((V ω, Aω)) // find set of paths

⋃P∈Π′ P = (V ′, A′)

9 for P = (V ′, A′) ∈ Π′ do10 Aω = Aω \ a ∈ AωT(k) | ∃k ∈ Kω s.t. AωT(k) ∩A′ 6= ∅11 Πω = Πω ∪Π′12 until Π′ = ∅ or AωT(k) = ∅13 while

⋃ω∈Ω Πω 6= ∅ do // SECOND PHASE

14 vopt = 015 (P , ω, S, W , K) = ((∅, ∅), 0, ∅, 0, ∅)16 for ω ∈ Ω do17 for P = (V ′, A′) ∈ Πω do18 valid = true19 (S, W , p, K) = (∅, 0, 0, ∅) // additional stations, budget, profit contr.,

trips20 if H ′ω = H ′ then W = W + Fcar // additional car required21 for k ∈ Kω s.t. ∃(it, jt′) ∈ A′ ∩AωT(k) do22 if k ∈ K ′ω then valid = false23 else24 p = p+ pk25 K = K ∪ k26 for (it, v) ∈ A′ \AωI do27 if i /∈ (S′ ∪ S) then // additional station(s) required28 S = S ∪ i29 W = W + Fi30 else if Cωit = 0 then valid = false31 if valid and W ′ + W ≤W then32 if vopt < Ψω p

W+ε then // cmp. efficiency (ε avoids division through 0)33 vopt = Ψω p

W+ε34 (P , ω, S, W , K) = (P, ω, S, W , K)35 else Πω = Πω \ P36 if vopt > 0 then // update solution37 Πω = Πω \ P38 H ′ω = H ′ω + 139 if H ′ < H ′ω then H ′ = H ′ + 140 K ′ω = K ′ω ∪ K41 S′ = S′ ∪ S42 W ′ = W ′ + W

43 for (it, jt′) ∈ A(P ) ∩ (AωW ∪AωC) do // update residual capacities44 Cωit = Cωit − 1

Algorithm 1: Path Heuristic

11

while the opening costs Fi are set to α + βCi where α and β are integers chosen randomly such thatα ∈ [100, 1000] and β ∈ [50, 100], respectively. The following parameter values have been consideredto obtain the final set instances: |Ω| ∈ 3, 5 with a randomly chosen probability for each scenario,|S| ∈ 10, 25, 50, |Kω| ∈ 10, 50, 100 for each scenario ω ∈ Ω, and Tmax ∈ 15, 30. The parameters ofeach trip k ∈ Kω are chosen as follows: origin ok and destination dk are chosen randomly while ensuringthat at least one potential station can be reached with the considered maximum walking time of tenminutes; start time sk randomly from 0, . . . , Tmax−1, end time ek randomly from sk . . . , Tmax, profitcontribution pk = 200(ek − sk), and a battery consumption of bk = ek − sk while using a rechargingrate of ρ = 10

3 , thus implying a recharging break of d0.3(ek − sk)e after trip k. Six instances have beencreated independently for each considered parameter combination. Table 1 summarizes the grid instancescreated and considered in our computational study, which are clustered into small, medium, and largeinstances according to their size. This clustering is based on our previous experiments [23] which showedthat the performance of the model strongly depends on the number of trips |K| as well as the numberof available cars H but considerably less on the number of stations |S|, scenarios |Ω| and the length ofthe planning period Tmax.

Table 1: Number of created grid instances (#) and number of considered stations (|S|), scenarios (|Ω|),trips per scenario (|Kω|) as well as time points in the planning period (Tmax) for each instance size.

# |S| |Ω| |Kω| Tmax

small 24 10 ∈ 3, 5 10 ∈ 15, 30medium 24 25 ∈ 3, 5 50 ∈ 15, 30

large 24 50 ∈ 3, 5 100 ∈ 15, 30

The following instance-independent parameter values have been used in our experiments. The pur-chase price of each car has been set to Fcar = 100 and four different values have been tested for thebudget W that are obtained as fractions of the overall investment costs necessary to construct all sta-tions and to purchase all available cars, i.e., we used W = w ·

(FcarH +

∑i∈S Fi

)for w ∈ 1

10 ,13 ,

12 ,

23,

which we refer to as budget fraction. The operating costs of the cars and stations are chosen relative totheir purchasing and constructing costs, respectively, i.e., φ = Fcar

100 , ϕ = Fi

70 . The maximum number ofavailable 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 testedour algorithm with H = 10dH

′

10 e + H+, for H+ ∈ 0, 10, i.e., we rounded up the number of availablecars 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 ofIntel Xeon E5-2670v2 machines with 2.5GHz, and a memory limit of 12GB has been set. Furthermore,an absolute time limit depending on the instance graph of 20 seconds (small instances), 8 hours (mediuminstances), or 24 hours (large instances) has been applied to each individual run. Our implementationwas 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 each experiment.

Results overview We first analyze the numbers of solved instances, average CPU-times and optimalitygaps for the three instance classes (small, medium, large). Thereby, we focus on the influence of thenumber 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 thegiven time and memory limits. The required CPU-time, however, drastically increases with increasinginstance size. Consequently, many of the large instances could not be solved to proven optimality. FromTables 2 and 3, we conclude that the running times, as well as the remaining optimality gaps increasewith an increasing number of scenarios. We also observe that instances with a larger number of cars aremuch more difficult for our algorithm. All these observations are supported by the fact that increasingthese parameters (number of scenarios, number of cars) lead to much higher numbers of variables andconstraints to be considered in the corresponding ILP instance. Conversely, the impact of increasing

12

Table 2: Numbers of solved instances (#OPT), average CPU-times (tavg) in seconds, and average opti-mality gaps (gapavg) in percent of the exact method (ILP) and the heuristic method. Optimality gapsgreater than zero are computed by considering only instances that were not solved optimally. Averagegaps of the heuristic method are relative to the best known solutions. The average CPU-times of theheuristic are only reported in the case of H+ = 0 since we did not run the heuristic again in the caseof H+ = 10, but only added 10 more cars as input parameter of the ILP. The CPU-times of the ILPdo not include those of the heuristic method. Results are grouped by the size of the instances, numbersof considered scenarios and numbers of available cars (relative to the number of cars used by the initialheuristic)

ILP Heuristic|Ω| H+ # #OPT tavg[s] gapavg[%] tavg[s] gapavg[%]

small 3 0 48 48 0.6 0.00 0.01 24.22

10 48 48 1.5 0.00 - 24.22

5 0 48 48 1.1 0.00 0.02 18.4610 48 48 3.4 0.00 - 18.46

med

ium 3 0 48 48 474.3 0.00 0.08 32.07

10 48 47 1 968.0 0.23 - 32.87

5 0 48 47 1 634.0 0.06 0.15 36.4210 48 45 5 445.4 0.11 - 37.29

large 3 0 48 19 68 175.1 1.20 0.28 40.18

10 48 7 81 937.3 2.28 - 45.02

5 0 48 13 73 373.6 10.15 0.55 41.5410 48 0 86 400.0 15.32 - 43.65

the number of considered time points (i.e., the value of Tmax) on the algorithm’s performance seemsinconsistent. To this end, we observe that the average CPU-times and remaining optimality gaps evendecrease when increasing the number of considered time points for medium sized instances with |Ω| = 3,as well as for large instances. We also conclude that even though most large instances could not besolved, the remaining optimality gaps seem acceptable from a practical perspective (typically less than2.5%) 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 manyscenarios. However, as is common in stochastic optimization, it will be crucial to select an appropriatebut sufficiently small set of scenarios when attempting to solve real world instances. In addition, it willbe important to use a good and not unnecessarily large estimation for the maximum number of cars.

Quality of the initial solution Supplemental to the overall performance analysis, it seems importantto analyze the quality of the solution computed by the heuristic algorithm introduced in Section 5. Theresults will provide insights whether a reasonable solution quality can be expected when solving verylarge instances with our heuristic that are too large for the developed ILP.

Figure 4 illustrates the gaps of the heuristic solutions relative to the best known solution values inpercent. We observe that the gaps increase with increasing instance size but also with decreasing budget.The latter observation can be explained by the fact that the heuristic algorithm might tend to open morestations instead of purchasing additional cars. Overall, it seems that the heuristic derives reasonablygood initial solutions for the ILP, but the development of additional improvement operators should beconsidered when attempting to solve very large instances with a rather limited budget. An alternativeoption with a significantly smaller development effort might be to incorporate some randomization com-ponents in Algorithm 1 and then repeatedly apply the resulting randomized heuristic. If the goal is toidentify stations maximizing the expected profit (or number of satisfied customer requests) without tightbudget restrictions, the current heuristic might be a viable option in practice.

Influence of the available budget As observed in the previous paragraph, the available budgethas a strong impact on the solution quality achieved by the heuristic algorithm. Thus, we also studyits influence on the performance of the overall algorithms composed of the successive application of

13

Table 3: Numbers of solved instances (#OPT), average CPU-times (tavg) in seconds, and average opti-mality gaps (gapavg) in percent of the exact and the heuristic method. Optimality gaps greater thanzero are computed by considering only instances that were not solved optimally. Average gaps of theheuristic method are relative to the best known solutions. The CPU-times of the ILP do not includethose of the heuristic method. Results are grouped by the size of the instances, numbers of consideredscenarios and numbers of considered time points.

ILP Heuristic|Ω| Tmax # #OPT tavg[s] gapavg[%] tavg[s] gapavg[%]

small 3 15 48 48 1.0 0.00 0.01 16.9

30 48 48 1.1 0.00 0.02 31.23

5 15 48 48 1.8 0.00 0.02 11.0830 48 48 2.7 0.00 0.03 25.85

med

ium 3 15 48 48 900.8 0.00 0.07 33.66

30 48 47 1 541.4 0.23 0.10 31.27

5 15 48 45 3 728.0 0.11 0.11 37.6630 48 47 3 351.3 0.06 0.18 36.04

large 3 15 48 8 79 195.6 2.13 0.24 42.72

30 48 18 70 916.7 1.44 0.33 42.48

5 15 48 5 81 062.7 14.81 0.50 42.3830 48 8 78 711.8 11.35 0.61 42.81

small medium large

020

4060

80100

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gaps

ofheurist

icsolutio

nsto

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ns(in

%)

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020

4060

80100

budget fraction w (in %)

gaps

ofheurist

icsolutio

nsto

best

know

nsolutio

ns(in

%)

Figure 4: Gaps of the heuristic solutions relative to the best known solutions for different instance classesand budget fractions.

the heuristic and the ILP. Figure 5 contains performance plots showing the fraction of instances solvedwithin 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 dependency of the performance on the availablebudget. The difficulty of an instance, however, does not strictly increase with increasing or decreasingbudget. Instead, cases with very low or high budget can be solved efficiently, while intermediate casesseem more difficult. While this trend is quite clear for small and medium instances, the small number ofsolved large instances prevents clear observations for the latter (with respect to solved instances). Withthe exception of imposing a very restricted budget (i.e., w = 0.1) to large instances, we also conclude

14

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Figure 5: Fraction of problem instances solved (to optimality) within the CPU-time limit in seconds andremaining optimality gaps in percent, with respect to the fraction w (i.e., available budget W relativeto the overall investment costs). Time limits were set to 20 seconds for problem small instances, 8 hoursfor medium problem instances and to 24 hours for large problem instances, respectively.

15

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: ViennaBesides testing our approach on grid instances, we also applied it to real-world instances based on thecity of Vienna provided to us by the Austrian Institute of Technology (AIT). In this dataset, the streetnetwork has been modeled based on OpenStreetMap data [33], which results in a graph representing thewhole city and contains 78 803 vertices and 198 642 arcs. A total of 693 potential stations are locatedat points of interest, e.g., supermarkets, parking places, and subway stations. Taxi trips (of a particularweek in spring 2014) have been used as an estimation of the car-sharing demand, since a taxi trip mightbe substituted by using a car-sharing service in case the latter is easily available. Origins, destinations,start and end times, as well as estimated battery consumptions (when performing the trip with anelectric vehicle) are associated to each of the 6 640 trips included in the data set1. Moreover, the profitcontribution of each trip has been set to 0.3e per minute. Note that this fee is oriented at the currentprices of the car-sharing operator car2go [11] in Vienna. The construction costs of the stations arebased 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 9 000- 64 000e as base-price, plus the cost of a slow charging point (17 000 - 26 000e) times the station’scapacity. The purchasing costs of a car where estimated at 15 000e including quantity discounts.

Seven scenarios were defined (one for each weekday) with probabilities of 0.15 for workdays, 0.13for Saturday and 0.12 for Sunday, respectively. Note that the planning horizon Tmax refers thereforeto exactly one day and a granularity of one hour (30 minutes, 15 minutes) was used, thus 24 (48, 96)time points are considered for each day and original start and end times have been rounded down (starttime) and up (end time) appropriately. A maximum walking time of five minutes has been used in ourexperiments. All tests were performed on the same hardware detailed in the previous section, with amemory limit of 28GB and a CPU-time limit of one week.

Besides applying the heuristic to this instance, we performed additional experiments with the exactapproach using a subinstance composed of the eight central districts of Vienna. As a densely populatedarea with scarce parking spaces and a relatively large number of inhabitants that do not own a privatecar, these districts seem to be an ideal region for testing an electric car-sharing system. The resultinginstance contains 13 311 vertices, 32 184 arcs, 201 potential stations and 1 060 trips. It turns out thatthe exact approach is able to solve this reduced, but still practically relevant instance. Figure 6 showsan optimal solution for a particular case in which the available budget was set to 5% of the overallinvestment costs (i.e., w = 0.05). Tables 4 and 5 summarize the obtained numerical results for the innerdistricts and whole Vienna, respectively.

From Table 4 we observe that typically all available cars are used in the exact solution. This confirmsthe previous observation, that the heuristic tends buy relatively few cars. All but one considered testcases could be solved to proven optimality. Surprisingly, the number of built stations is usually largerthan the number of purchased cars. From the average numbers of accepted trips, we further concludethat each car typically performs several trips in each scenario. Furthermore, we conclude that usingat most one third of the maximum investment costs (i.e., w = 0.33) seems sufficient to obtain a mostprofitable solution since the numbers of built stations, purchased cars, and performed trips do not changefor larger values of w.

From Table 5 we conclude that the solutions produced by the heuristic do not seem to change alot with increasing granularity of the planning period. As opposed to the exact results for the innerdistricts, providing more budget than 33% of the total investment costs still yields a significant increasein the numbers of open stations, purchased cars and accepted trips. This observation may stem fromdifferent instance characteristics when considering also the outer parts of Vienna or from the fact thatthe heuristic does not make best use of the given budget. Overall, we also observe that a relativelylarge portion of the trips are accepted in the heuristic solution if enough budget is available and that theaverage trip-to-car ratio per scenario typically is between one and two.

1The dataset actually contains 37 965 trips, from which we took the ones for which at least one potential station existswithin the chosen maximum walking distance. Furthermore we excluded trips that do not start and end at the same day,respectively.

16

Table 4: Results of the exact method on instances based on the inner districts of Vienna, grouped by thenumber of available cars (relative to the number of cars used by the initial heuristic) and the availablebudget as fraction w of the overall investment costs. Besides the numbers of open stations (|S′|) andpurchased cars (H ′), we also report the average number of accepted trips per scenario (K ′) as well asthe CPU-time (t) in seconds and optimality gaps (gap) in percent of the exact method (ILP) and theheuristic method. The gaps of the heuristic method refer to the ratio of the objective values obtainedfrom the heuristic solutions relative to the best known objective values.

ILP HeuristicH+ w |S′| H ′ K ′ t[s] gap[%] t[s] gap

0

0.10 30 10 76 109 729.0 0.00 25.2 61.840.33 49 10 80 4 571.9 0.00 25.2 67.850.50 48 10 80 4 486.5 0.00 25.3 61.840.66 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 61.840.33 70 20 107 57 908.0 0.00 25.2 67.860.50 70 20 107 51 157.6 0.00 25.3 61.210.66 70 20 107 56 762.6 0.00 25.2 64.70

Table 5: Heuristic results of instances based on the data of whole Vienna, grouped by the granularityof the planning period (Tmax) 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 perscenario (K ′) as well as the CPU-time (t) in seconds for solving the heuristic are given.

Tmax w |S′| H ′ K ′ t[s]

24

0.10 88 40 69 9 076.20.33 258 173 320 9 087.00.50 364 273 459 9 101.80.66 461 300 513 9 088.8

48

0.10 81 34 52 11 448.30.33 248 163 301 11 454.80.50 353 236 429 11 469.60.66 450 299 478 11 497.4

96

0.10 88 36 59 10 725.60.33 245 156 303 10 734.40.50 359 229 428 10 750.70.66 454 264 465 10 761.9

17

We also provide summarized economic results of the instances based on the inner districts of Viennagiven in Table 6, which may act as an exemplary basis for decision-makers. Thereby, we assume anaverage asset depreciation range of eight years and thus a depreciation rate of 12.5% per year, as wellas a residual value of the capital goods of 5% of the investment costs, for the computations. Moreover,we assume that an operator can fully take advantage from the depreciation by reducing her/his earningsbefore taxes from other investment projects. Notice that the results imply that a stand-alone-investmentin an electric car-sharing system would be unprofitable in the investigated case, without the latterassumption. We further remark that we used a static approach to obtain the reported results and thatthey also strongly depend on the model assumptions (given in Section 3) in practice. However, theresults indicate that the implementation of an electric car-sharing system in the inner districts of Viennacan be profitable with respect to the underlying assumptions, since the average expected payoff timesare significantly lower than the assumed average asset depreciation range. This may offer opportunitiesfor established companies to act as first mover and to gradually expand the operational area, whileconsidering the expected decreasing costs of electric mobility in the years to come. Furthermore, onecan expect the results to improve if more available cars are considered in the computations, as signifiedin the previous paragraph. Moreover, we observe similar results of the exact method and the heuristicmethod with respect to the relative measurements of profitability (i.e., average expected payoff time andrate of return), and therefore conclude that the heuristic method delivers useful results from an economicperspective, despite the relatively large (optimality) gaps.

Table 6: Summarized economic results of the exact method (ILP) and the heuristic method on instancesbased on the inner districts of Vienna. Besides the average investment costs and yearly operating costswe report the average expected profit per year, payoff time and rate of return per year.

ILP Heuristicavg. investment costs 5 560 125 e 1 909 000 eavg. operating costs, p.a. 67 648 e 23 226 eavg. expected profit, p.a. 136 683 e 48 373 eavg. expected payoff time 6.2 years 6.3 yearsavg. expected rate of return, p.a. 5.5 % 4.8 %

8 Conclusions and OutlookIn this article, we introduced and studied a stochastic optimization problem that aims to solve thestrategic optimization problem of determining optimal locations for charging stations of (ad-hoc) electriccar-sharing systems. We observed that though such systems may help to overcome important (environ-mental) challenges arising in cities, there is little scientific research on the use of electric cars withingcar-sharing systems. After stating a couple of practically relevant and important assumptions (such asavailability of an appropriate demand forecast) we gave a formal definition of the resulting two-stagestochastic optimization problem.

The problem was modeled as an integer linear program in which a set of time-expanded locationgraphs is used to track each car’s position during the planning period. We also proposed a heuristicalgorithm that is based on the idea of solution construction by iteratively adding profitable paths thatcorrespond to routes of individual cars. The heuristic has been used as a standalone algorithm for verylarge scale instances as well as to provide an initial feasible solution for the subsequent application ofthe time-dependent ILP model.

A computational study on a set of grid-graph-based test instances was performed to analyze theinfluence of different parameters on the overall performance. The obtained results show that the difficultyof instances increase noticeably with an increasing number of available cars. The used granularity of theplanning period, however, does not seem to have a large impact on the required solution time. We alsoshowed that instances with a tightly constrained or almost unconstrained budget are often relatively easyto solve, while those in-between (i.e., where the budget is still rather constraining but already allows tobuild a significant number of stations and to purchase several cars) seem more difficult.

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Finally, we performed a case study on real-world instances from Vienna. It turns out that thedeveloped exact approach is suitable for solving instances obtained for eight central districts, but cannotbe applied to instances in which the street network models the whole city of Vienna, where we successfullyapplied the heuristic algorithm. The reasonably chosen locations for charging stations on these real-world instances confirm the suitability of the proposed optimization problem. Moreover, we consideredan economic perspective and showed that the implementation of an electric car-sharing system in theinner district does not seem profitable as a stand-alone-investment, but can be profitable next to otherprojects, which might offer opportunities for established companies to act as first mover in that businessarea.

Several possibilities for future research can be derived from our results. From a computationalperspective, it might be worth to develop sophisticated decomposition methods that are likely to yieldan exact algorithm with a significantly better performance as the current one. Such an algorithm mightbe suitable for large-scale real-world instances such as those obtained from considering the whole cityof Vienna. Alternatively, one might consider the development of metaheuristic approaches that mightbe able to derive better solutions than the greedy heuristic proposed in this article. Besides algorithmicimprovements, relevant research directions include the development of models that also consider therelocation of cars by the operator, or that relax the assumption that cars need to be fully rechargedbefore every trip.

AcknowledgementsThe authors thank their project partners from the Austrian Institute of Technology (AIT) for creatingthe real-world instance from Vienna. This work is supported by the Joint Programme Initiative UrbanEurope under the grant 847350 and by the Austrian Science Fund (FWF) under grant I892-N23. Thesesupports are greatly acknowledged.

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0 0.5 1 km

open stations

closed stations

Figure 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 minutes that users are expected to be willing to walkto/from a station are indicated by the large icons and circles, respectively, whereas the closed stationsi ∈ S \ S′ are represented by the small icons. Note that the circles representing the areas covered byopen stations are only graphical approximations and the exact set of nodes reachable within five minuteshas been used in the computation.

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Appendix A List of sets, variables and parameters

Table 7: Tabular description of the notation used throughout the paper.

Group Notation Description

Inpu

tpa

rameters

A Arcs of input graph (street network)Bmax Battery capacity of each carbk Battery consumption (overestimated) of trip k ∈ Kdk Destination dk ∈ V of trip k ∈ KCi Capacity of a station i ∈ S (maximum number of charging slots)∆k Duration ∆k = ek − sk of trip k ∈ Kek End time ek ∈ T of trip k ∈ KFi Construction costs of potential station i ∈ SFcar Purchasing costs per carH Number of available carsS Set of potential stationsϕi Operating costs of station i ∈ Sφ Operating costs per purchased carG Input graph (street network) G = (V,A)K Set of potential trips defined as the union of trips per scenario, i.e., K =

⋃ω∈ΩK

ω

each trip k ∈ K is given as tuple (ok, dk, sk, ek, bk, pk) representing origin ok, destination dk,start time sk, end time ek, battery consumption bk, and profit contribution pk

Kω Set of potential trips in scenario ω ∈ ΩNi 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)ok Origin ok ∈ V of trip k ∈ Kpk Profit contribution of trip k ∈ KΨω Probability of scenario ω ∈ Ωρ Rate of recharge (per time unit)sk Start time sk ∈ T of trip k ∈ KT Planning period T = 0, . . . , TmaxTmax End of planning periodV Nodes of input graph (street network)W Available budgetΩ Set of scenarios with probabilities Ψω for each ω ∈ Ω

Solution

end(k) End station of accepted trip k ∈ K ′H ′ Purchased carsK ′ Accepted tripsK ′h(ω) Sequence of trips (in temporal order) performed with car h in scenario ω ∈ ΩS′ Built stationsstart(k) Start station of accepted trip k ∈ K ′W ′ Used budget

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Table 7 – continued from previous pageGroup Notation Description

Variables fha Whether or not car h ∈ 1, . . . ,H travels along arc a ∈ Aω in scenario ω ∈ Ω (second stage)

xk Whether or not trip k ∈ K is accepted (second stage)xhk Whether or not trip k ∈ K is assigned to car h ∈ 1, . . . ,H (second stage)yi Whether or not station i ∈ S is built (first stage)zh Whether or not car h ∈ 1, . . . ,H is purchased (first stage)

Tim

e-exp.

grap

hs

Aω Arcs of time-expanded location graph of scenario ω ∈ ΩAωI Set of initial allocation arcs of time-expanded location graph in scenario ω ∈ ΩAωW Set of waiting arcs of time-expanded location graph in scenario ω ∈ ΩAωC Set of final collection arcs of time-expanded location graph in scenario ω ∈ ΩAωT Set of travel arcs of time-expanded location graph in scenario ω ∈ Ωrω Artificial root node of time-expanded location graph of scenario ω ∈ Ωsω Artificial sink node of time-expanded location graph of scenario ω ∈ ΩGω Time-expanded location graph Gω = (V ω, Aω) of scenario ω ∈ ΩV ω Nodes of time-expanded location graph of scenario ω ∈ Ω

Heu

ristic

Aω Arc set of subgraph GωCωit Residual capacity station i ∈ S at time t ∈ T in scenario ω ∈ ΩGω Subgraph Gω = (V ω, Aω) of time-expanded location graph Gω, ω ∈ Ω,

obtained by removing all trip arcs corresponding to already covered tripsH ′ω Number of used cars in scenario ω ∈ Ω in current (partial) solutionK Set of trips in current pathK Set of trips in currently most efficient pathΠω Set of non-dominated paths in scenario ω ∈ ΩΠ′ Temporary set of non-dominated pathsP Currently considered pathP Most efficient path in current iterationp Total profit contribution of current pathS Set of additional stations required for current pathS Set of additional stations required for most efficient pathV ω Node set of subgraph Gωvopt Efficiency value of currently most efficient pathW Required budget for current pathW Required budget for currently most efficient path

Gen

eral

δ+(it) Set of outgoing arcs of vertex it ∈ V ω in time-expanded location graph Gω, ω ∈ Ωδ−(it) Set of ingoing arcs of vertex it ∈ V ω in time-expanded location graph Gω, ω ∈ Ωfh[A′] Sum of flow variables over all arcs in subset A′ ⊂ Aω, i.e.,

∑a∈A′ f

ha

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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