**3. Optimization Model**

The focus of this article is on the round-trip carsharing service, which received relatively less attention in the research community [52]. Hence, a novel approach for round-trip carsharing fleet placement is proposed. There are three benefits that the approach offers. The first benefit is that automation of the fleet placement process removes the need for traditional manual allocation. The second benefit is the higher placement precision with the inclusion of a Geographic Information System (GIS). Finally, the third benefit is in proposing an approachable fleet managemen<sup>t</sup> problem to the research community that may also be beneficial to similar applications.

The proposed approach emphasizes the user coverage and ease of access with a constraint of being applicable to the real-world scenario by utilizing the two components mentioned below. These two goals are of real concern in practice and are often expressed by experts in this area.


In this section, the graph instance used in this article is first defined for the FPP since the graph instance is closely related with the problem definition. Second, the Fleet Placement Problem (FPP) model is formulated. Finally, FPP is proven that it is an NP-hard in a strong sense.

#### *3.1. Graph Instance Definition*

The street map can be modeled as an undirected weighted graph to represent users' ability to walk on the streets in both directions where they need to pick up or leave the shared vehicle (see Figure 1). The street-level graph is modeled as follows:

$$G = (V, E, P, W).$$

The set of nodes *V* is composed of two subsets: *V* = *S* ∪ *B*, where *S* is the subset of street nodes (i.e., nodes on roads and streets), while subset *B* contains buildings. Both *S* and *B* are of type node, hence they are naturally members of vertices in *G*.

Buildings contain users. A weight *pi* ∈ *P* associated with each node *bi* ∈ *B* corresponds to the estimation of the number of people living in this building. The weight for each node in *S* is set to zero, making the assumption that target users are only located in buildings. This is similarly to the study taken by Daniels and Mulley [53]).

The set of edges *E* consists of two subsets: *E* = *R* ∪ *L*, where *R* is a set of streets/roads, and *L* is a set of links connecting residential buildings to nearby streets.

Each edge (*<sup>u</sup>*, *v*) ∈ *R* is valuated by a weight, *wi* ∈ *W*, representing the walking distance from *u* to *v* where *u*, *v* ∈ *S*.

Each edge (*<sup>u</sup>*, *b*) ∈ *L* where *u* ∈ *S* and *b* ∈ *B* is valuated by 0. In other words we consider the distance between the building and its nearest adjacent street is considered negligible. This process is called "snapping" and is common in every routing service where the starting point is first projected on a road before starting to build a route [54].

**Figure 1.** An illustration of a graph instance.

#### *3.2. Fleet Placement Problem*

The concept of a virtual station is created in this work, In the FPP model, carsharing stations are placed on the streets. A station is a virtual area on the city map defined by two elements, a center point, and a radius. Figure 2 illustrates this concept. A car is placed on a road which depicts a center point. A circle represents a radius of that center point. These two elements constitute the virtual station. A car can be picked up and returned to anywhere in that circle (station area). The coverage of a station is determined by the given maximum walking distance illustrated as a green line in the figure.

Due to the round-trip nature of the service, each car taken from the station needs to be returned to this station after completing the trip. The typical customers are people in residential areas (as shown in Figure 2) covered by the carsharing stations, who walk to pick up the nearest vehicle. Under the aforementioned assumptions and constraints, FPP can then be formulated.

**Figure 2.** An illustration of a virtual area of a station on streets.
