## 2 Problem Definition
If we denote by \(\Gamma\) the set of lot classes and by \(\gamma(\ell)\in\Gamma\) the class of a lot \(\ell\), we introduce coefficients \(\mathrm{LB}^{7}_{d}\), \(\mathrm{UB}^{7}_{d}\) to, respectively, represent the lower and upper bound on the number of lots of class \(\gamma\in\Gamma\) that can be deployed in each district \(d\in D\). After having introduced all the required system elements, we can state the problem that we intend to model. **Definition 1** (The E-Scooter Parking Lot Rental Problem-ESPLR). : _Given the set of districts \(D\), each partitioned into a set of subdistricts \(S(d)\), the set of parking lots that can be potentially rented \(L(d,s)\), the bounds on the number of lots, car places subtracted and lot types \(\mathrm{LB}^{\mathrm{TOT}}_{ds}\), \(\mathrm{UB}^{\mathrm{TOT}}_{ds}\), \(\mathrm{UB}^{6}_{ds}\), \(\mathrm{UB}^{7}_{d\ell}\), \(\mathrm{UB}^{7}_{d\ell}\), the numbers \(c_{ds\ell}\) of car places occupied by lots, the number of spaces \(n_{ds\ell}\) hosted by lots and the values \(v_{ds\ell}\) of the lots, the E-Scooter Parking Lot Rental Problem (ESPLR) is that of choosing the parking lots that are rented in each subdistrict, respecting all the bounds, with the aim of maximizing the total value of the rented lots._  
The ESPLR problem can be modeled as a Binary Linear Programming problem, including the decision variables, feasibility constraints and objective function that follow. The first component of the problem is represented by binary decision variables \(x_{ds\ell}\in\{0,1\}\), for each district \(d\in D\), subdistrict \(s\in S(d)\) and lot \(\ell\in L(d,s)\), which assume value according to the following rules:  
\[x_{ds\ell}=\left\{ \begin{array}{ll}1&\quad\text{if lot $\ell$ in subdistrict $s$ of district $d$ is rented}\\ 0&\quad\text{otherwise}\end{array} \right. \tag{1}\]  
Then, we introduce a set of constraints for expressing the bounds on the total number of parking places that can be rented in each subdistrict:  
\[\mathrm{LB}^{\mathrm{TOT}}_{ds}\leq\sum_{\ell\in L(d,s)}n_{ds\ell}\cdot x_{ds \ell}\leq\mathrm{UB}^{\mathrm{TOT}}_{ds}\qquad\forall d\in D,s\in S(d) \tag{2}\]  
Note, that in the previous constraints, a decision variable \(x_{ds\ell}\) is multiplied by the corresponding number of parking places \(n_{ds\ell}\) that the lot \(\ell\) includes. Then, we introduce a set of constraints for expressing the upper bound on the total number of car places that can be subtracted by e-scooter parking lots in each subdistrict:  
\[\sum_{\ell\in L(d,s)}c_{ds\ell}\cdot x_{ds\ell}\leq\mathrm{UB}^{\mathrm{c}}_ {ds}\qquad\forall d\in D,s\in S(d) \tag{3}\]  
Note that, in the previous constraints, the number of subtracted places is obtained by the multiplication of a decision variable \(x_{ds\ell}\) with the coefficient \(c_{ds\ell}\) expressing the number of car places occupied by the e-scooter parking lot \(\ell\). Eventually, we introduce constraints for modeling the lower and upper bounds on the numbers of lots of class \(\gamma\in\Gamma\) that can be deployed in each district:  
\[\mathrm{LB}^{7}_{d}\leq\sum_{s\in S(d)}\ \sum_{\ell\in L(d,s):\ \gamma(\ell)= \gamma}x_{ds\ell}\leq\mathrm{UB}^{7}_{d}\qquad\forall d\in D,\gamma\in\Gamma \tag{4}\]We remark that, here, the second summation has the aim of selecting only those decision variables associated with lots of district \(d\) whose class coincides with the class \(\gamma\) of the constraint. The objective function provides for maximizing the total value of rented lots, considering all districts and subdistricts:  
\[\max\sum_{d\in D}\sum_{s\in S(d)}\sum_{\ell\in L(d,s)}v_{ds\ell}\cdot x_{ds\ell} \tag{5}\]  
The overall optimization model, obtained by combining all the variables, constraints and objectives (1)-(5) is written as:  
\[\max\sum_{d\in D}\sum_{s\in S(d)}\sum_{\ell\in L(d,s)}v_{ds\ell} \cdot x_{ds\ell} \text{(BLP-ESPLR)}\] \[\quad\  
end of the solution construction phase.