# Optimization of Carsharing Fleet Placement in Round-Trip Carsharing Service
## 2 Related Work
### 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.  
[PERSON] and [PERSON] 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. [PERSON] used MCLP as a base model for placing healthcare facilities based on the demographic data in the regions [7]. [PERSON] 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 [PERSON] et al. who proposed a bi-level version of the hub interdiction problem (also another variant of MCLP) [9]. MCLP was also extended to its multi-objective. [PERSON] et al. proposed a MCLP with two objectives which were facility cost and proximity minimization [10]. [PERSON] 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]. [PERSON] 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.