**1. Introduction**

It is undeniable that efficient managemen<sup>t</sup> of transportation has become one of the major problems in cities across the globe due to its impact on the environment and quality of life. Carsharing is one of many means of transportation nowadays and has received positive support from communities and governments. Its success can be seen in several countries, such as Germany, which has the biggest carsharing market in Europe with over 2 million registered users, 170 service providers, and over 16,000 vehicles available in 740 cities [1,2]. Coupling with the increasing awareness in environmental problems, the concept of green mobility is also promoted through the electrical carsharing service [3]. For instance, it has been highlighted that cars are used for transportation more than trains and planes in Germany and that carsharing positions itself is an intermediate mean to fill the gap between public transport and personal cars [4]. Another success was reported in

**Citation:** Changaival, B.; Lavangnananda, K.; Danoy, G.; Kliazovich, D.; Guinand, F.; Brust, M.R.; Musial, J.; Bouvry, P. Optimization of Carsharing Fleet Placement in Round-Trip Carsharing Service. *Appl. Sci.* **2021**, *11*, 11393. https://doi.org/ 10.3390/app112311393

Academic Editor: Juan-Carlos Cano

Received: 16 October 2021 Accepted: 24 November 2021 Published: 1 December 2021

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the United Kingdom where the governmen<sup>t</sup> provided support to extend the user base to 600,000 individuals by 2020 to reduce traffic and parking problems [2].

The carsharing model can be divided into free-floating and station-based [2]. Freefloating carsharing offers the highest degree of flexibility to the users. They pick up the nearby vehicle to start a trip and drop it off anywhere in the city to end the trip. However, flexibility comes with a high operational cost for the company, which needs to maintain a high density of vehicles even in low-demand areas of the city in order to cope with low levels of utilization. Additionally, vehicles that end up in low demand areas need to be reallocated. An example of well-known free-floating carsharing companies are SHARE NOW (which is the merge of Car2Go and DriveNow) [1].

In station-based carsharing, fleet vehicles are stationed in densely populated areas of the city and need to be returned to one of these locations after completing the trip (one-way) or to the same pick up area (round-trip). As a result, station-based services are less flexible, with an advantage of easier implementation and management. Station-based services require fewer cars, as carsharing operators can place vehicles in densely populated and high-demand zones of the city only, and no fleet relocation is typically required. An example for easier implementation and managemen<sup>t</sup> is taken from bike-sharing. The bicycles need to be relocated everyday to maintain the service function, which is relatively easy because of their sizes. However, car relocation is more difficult and more expensive, which is an even bigger burden to carsharing operators. In addition, with the trend of electronic cars, a charging station is easier to implement in the station-based services due to having fewer stations to implement than the free-float services.

Therefore, a satisfactory solution associated with carsharing is multifarious. The work in this article is concerned with maximum user coverage and ease of access to the service (i.e., shortest distance to a station and flexibility in returning). It comprises the following:


In this work, we propose a new method attempting to optimize the fleet placement in the station-based round-trip, will will be the first to tackle fleet allocation in roundtrip carsharing. The model of this problem is called fleet placement problem (FPP). Fleet placement is really tedious and is usually performed manually by experts and hence is prone to errors due to lack of precision. The proposed methodology aims to maximize customer coverage, while minimizing the maximum walking distance between customers and the nearest vehicle. These two objectives are in conflict, thus, resulting in a bi-objective problem. Unlike previous solutions, the proposed model incorporates highly detailed street-level map data containing footprints of the buildings. The contributions proposed in this work are: mathematical formulation of the novel fleet placement problem (FPP) and its NP-hardness proof, correlation analysis of the two problem objectives, and the comparison of results between the manual placement and state-of-the-art heuristic and metaheuristic algorithms.

The remainder of this paper is organized as follows. Related work on fleet location for carsharing services and similar location problems are presented in Section 2. The formulation of the FPP problem and the methodology to solve it are detailed in Section 3. In Sections 5 and 6 results from executions on real city instances are presented and discussed. Finally, conclusions and perspectives are provided in Section 7.

#### **2. Related Work**

In this section, the state-of-the-art on fleet placement and location problems, shared fleet placement, and the optimization methods used to address are analyzed.

#### *2.1. Fleet Placement and Location Problems*

In the following, the similarities of the fleet location problem with two classical optimization problems, i.e., the maximal covering location problem (MCLP) and the facility location problem (FLP) are discussed. These two problems and the proposed fleet placement problem (FPP) can be reduced to the set covering problem as shown in the proof in Section 3.

Church and ReVelle proposed the maximal covering location problem (MCLP) in 1974 [5] for facility and emergency siting. The objective is to maximize the partial coverage with a number of facilities, where each facility has a fixed coverage distance. MCLP is shown to be NP-hard, which means it becomes intractable and cannot be solved in an acceptable time by exact methods when the size of an instance is large [6]. MCLP has been applied in many real-world problems. Seargeant used MCLP as a base model for placing healthcare facilities based on the demographic data in the regions [7]. Schmid et al. formulate their ambulance siting problem as an MCLP with the integration of patients' data and traces of taxis in Vienna to estimate the traveling time to reach the patient [8]. Another example in telecommunication is from Ghaffarinasab et al. who proposed a bilevel version of the hub interdiction problem (also another variant of MCLP) [9]. MCLP was also extended to its multi-objective. Xiao et al. proposed a MCLP with two objectives which were facility cost and proximity minimization [10]. Kim et al. solved another bi-objective version of the MCLP where the aim was to maximize primary and backup coverage (overlapping coverage for reliability) [11]. Malekpoor et al. formulated the problem of electrification in a disaster relief camp as finding a set of locations to reduce the project cost and increase the share of systems between sites [12].

In the facility location problem (FLP), the objective is to find locations to place facilities to supply stores, while minimizing the maximum cost (p-center) or the average cost (p-median) [13]. In this problem, one constraint is to have all the stores covered while one store can be covered by only one facility [14]. One of the many interesting applications of the FLP is shown in [15] where they utilized the spatial information and studied the difference between the optimal facilities locations and the current ones. Another application is in siting rescue boat locations. It was modelled as a multi-objective problem which considers not only the response time to the incidents, but also the operating cost and working hours [16]. FLP was also used in telecommunication to find the location of GSM antennas as shown in [17,18].

These two problems are highly related to our fleet placement problem (FPP). The similarity between FPP and MCLP is that they both try to maximize the partial coverage, with a constraint of fixed coverage distance and fixed number of facilities. Meanwhile, the FLP objective is to minimize the maximum operating cost, which is well aligned with FPP, with a second objective, which is to minimize the maximum walking distance. Therefore, FPP can be seen as a combination of these two problems.

#### *2.2. Shared Fleet Placement*

Shared fleet placement can be formulated into MCLP or FLP (especially in round-trip carsharing service). However, there are other factors to be considered. In previous works on MCLP and FLP, they already have a list of preferred locations. These possible sites are evaluated by considering convenience factors such as parking cost, the proximity to essential facilities, and accessing time as presented in [19,20]. The solution was then a combination of selected sites to maximize user coverage. In fact, they are very similar to the facility location problem. Kumar and Bierlaire evaluated potential stations by the distance between the station and other facilities such as hospitals and train stations. They also had access to historical data to make a decision on where to place the station, which is not available in most cases [20]. Another popular approach is to locate the fleet by user demands [21–23]. Boyacı et al. proposed an optimization model to maximize the user coverage based on the demand and predicted destinations [21]. On the contrary, Lage et al. studied a method to identify the potential of city districts in a station-based one-way trip

scenario where the demands were estimated from the taxi trips and customer profiles in Sao Paulo, Brazil [22]. Lastly, Schwer and Timpf proposed an idea for locating the fleet in round-trip carsharing by combining both user demands and proximity to other mean of transportation and other facilities into a model and utilized the open source data available from the governmen<sup>t</sup> [23].

There are also works which focus on electric carsharing fleets, reflecting the increasing awareness of environmental issues and benefiting from governmental support. The electric carsharing fleet is more complicated than its fossil-fuel counterpart considering the additional constraints for battery/electrical load managemen<sup>t</sup> of the car. Çalik and Fortz proposed a model for a one-way electric carsharing service which considered previously mentioned factors [24]. Since charging is very important in electric carsharing service, Jiao et al. [25] formulated their model to consider a situation where the user changes the drop off station. The charging station location optimization was presented by Brandstätter et al. [26] where the authors based the location on the source and destination of trips in both simulation and Vienna. Another example was proposed by Yıldız et al. [27] which consider a more realistic case where demand was stochastic and capacitated charging stations. In addition, the shared fleet placement is also studied in the bikesharing community where station locations and bike stocking are highly important as well [28–32].

#### *2.3. Existing Resolution Approaches*

Several algorithms were proposed to solve the aforementioned problems, ranging from exact methods, to heuristic and metaheuristic algorithms. Exact methods guarantee optimality, however, once the size of an NP-hard problem is too large, such methods (e.g., branch-and-bound, exhaustive search) cannot find solutions in reasonable time. In contrast, heuristic algorithms (e.g., greedy algorithms) are problem specific methods that permit to obtain an approximate solution in reasonable time. Finally, metaheuristic algorithms (e.g., genetic algorithm or simulated annealing), are general purpose algorithms, which can lead to very satisfactory solutions. A true benefit is their acceptable execution time, which for middle to large size instances is several orders of magnitude smaller than for exact methods [33].

For generic location problems such as MCLP and FLP, exact methods are usually used [7,8,12,16,34–36]. It is important to note that the problem instances tackled in the reported articles were of limited size. In fact, Zarandi et al. [36] reported that IBM CPLEX [37] cannot handle a problem with a large size of input (e.g., a city). Hence, once the problem size is too large, heuristic and metaheuristic algorithms are usually employed. Church and ReVelle [5] first compared two variations of heuristic algorithms (which add one facility location one at a time) and a branch and bound algorithm. The next attempt in solving MCLP was using a Lagrangian heuristic algorithm, which is a combination of the Lagrangian Relaxation approach and a greedy method [38]. Heuristic algorithms are still being used nowadays as shown in [39] to solve the FLP problem. Lastly, several works reported on the efficiency of metaheuristics in solving FLP and MCLP. Tabu Search (TS), Simulated Annealing (SA), Variable Neighborhood Search (VNS), and Genetic Algorithms (GA) were also considered in solving MCLP [36,40–43]. Metaheuristics were not only used to solve single-objective versions of these problems but also multi-objective ones. Xiao et al. [10] employed a Multi-Objective Evolutionary Algorithm (MOEA) to solve the bi-objective MCLP, which focused on facility cost and proximity minimization using a specific encoding scheme and dedicated operators. Kim and Murray [11] solved the bi-objective reliability-focused MCLP where it aimed to maximize primary and backup locations coverage with a heuristic algorithm and a Multi-Objective Genetic Algorithm (MOGA). Karasakal and Silav [44] utilized the crowding distance function from the Nondominated Sorting Genetic Algorithm II (NSGA-II) [45] in the Strength Pareto Evolutionary Algorithm II (SPEA2) [46] and reported that the new algorithm outperformed the original NSGA-II and SPEA-II. Ranjbartezenji et al. [47] proposed their modified version of NSGA-II and used it to solve bi-objective MCLP.

In shared fleet placement, the most common approach is to model it as a singleobjective problem (weighted sum) and to rely on exact solvers such as CPLEX or MATLAB [20,21,25,48,49]. Several path-based heuristic algorithms were proposed [24,26]. In fact, due to the problem complexity, heuristic and metaheuristic algorithms have been attracting more attention recently [50]. Another approach is to determine the fleet locations through agent-based simulation [51].

To date, metaheuristic algorithms have not been applied to shared fleet placement problem. Previous works mainly either consider small-size instances of the problem (at most 800 potential locations by curation) and apply exact methods, or propose heuristic algorithms. The location curation is normally conducted by field experts and is essential to facilitate the applications of exact methods and proposed heuristic algorithms, but it can be very time consuming and prone to error. Therefore, we aim to eliminate the curation process (to be fully automated) which in turn, leaves decision makers to consider over 100,000 locations (in cities like Munich). General exact methods and previously proposed heuristic algorithms are too computationally expensive to apply. Therefore, metaheuristic algorithms are a suitable candidate to solve such big instances of fleet placement problem efficiently.
