## 3 Model and Methodologies
### Rebalancing Phase  
#### 3.2.1 Multi-Objective MIP Model for PDVRP of Bike Rebalancing  
In this section, we develop a multi-objective MIP model for the PDVRP. The PDVRP is an extended model in which pick-ups and deliveries are performed while management vehicles move. First, we define the parameters and decision variables in Table 1 as follows. In this study, we simultaneously consider two objectives. The first is to minimize the sum of the differences between target inventory \(T_{i}\) and final inventory \(y_{i}\) after the rebalancing work for all stations \(i\in N\) as shown in Equation (3):  
\[f_{1}(x)=\sum_{i=1}^{n}(s_{i}^{+}+s_{i}^{-}) \tag{3}\]  
The other objective is to minimize the maximum traveling distance among the travel routes of the management vehicles as shown in Equation (4):  
\[f_{2}(x)=D^{max} \tag{4}\]  
Then, the MIP model for the multi-objective PDVRP is formulated as follows:  
\[\text{Min} f_{1}(x) \tag{5}\] \[f_{2}(x)\] \[\text{s.t.} \sum_{k=1}^{p}x_{ik}=1\quad\forall i\in N\] (6) \[x_{0k}=1\quad\forall k\in V\] (7) \[x_{(n+1)k}=1\quad\forall k\in V \tag{8}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Notations** & **Description** \\ \hline
**Parameters** & \\ \hline \(N\) & Set of stations where \(n=|N|\) \\ \(V\) & Set of vehicles where \(v=|V|\) \\ \(C\) & Vehicle capacity \\ \(l_{i}\) & initial inventory at station \(i\in N\) \\ \(T_{i}\) & Target inventory level at station \(i\in N\) \\ \(p_{ik}\) & Number of bikes that vehicle \(k\in V\) picked up from station \(i\in N\) \\ \(D_{ij}\) & Distance between station \(i\in N\) and station \(j\in N\) \\ \hline \hline \multicolumn{2}{l}{**Decision Variables**} \\ \hline \(x_{ik}\) & 1 if vehicle \(k\in V\) visits station \(i\in N\), 0 otherwise \\ \(z_{ijk}\) & 1 if vehicle \(k\in V\) moves from station \(i\in N\) to station \(j\in N\), 0 otherwise \\ \(t_{ijk}\) & No. of bikes that vehicle \(k\in V\) moves from station \(i\in N\) to station \(j\in N\) \\ \(y_{i}\) & Final inventory level at station \(i\in N\) \\ \(u_{ik}\) & Instrumental variable to prevent sub-tour \\ \(s_{i}^{+}\) & Positive inventory deviation from target inventory at station \(i\in N\) \\ \(s_{i}^{-}\) & Negative inventory deviation from target inventory at station \(i\in N\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of notations and decision variables. \[\sum_{j=1}^{n}z_{jik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V \tag{9}\] \[\sum_{j=1}^{n}z_{0 jk}=1\quad\forall k\in V\] (10) \[\sum_{j=1}^{n}z_{j(n+1)k}=1\quad\forall k\in V\] (11) \[z_{0(n+1)k}=0\quad\forall k\in V\] (12) \[z_{(n+1)0k}=1\quad\forall k\in V\] (13) \[\sum_{j=1}^{n}z_{ijk}\leq 1\quad\forall i\in N,k\in V\] (14) \[\sum_{k=1}^{v}z_{ijk}\leq 1\quad\forall i\in N,j\in N\] (15) \[x_{ik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V\] (16) \[u_{0k}=0\quad\forall k\in V\] (17) \[\sum_{j=1}^{n}z_{jik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V \tag{9}\]0 (18) \[\sum_{j=1}^{n}z_{jik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V \tag{9}\]1 (19) \[\sum_{j=1}^{n}z_{jik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V \tag{9}\]2 (20) \[\sum_{j=1}^{n}z_{jik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V \tag{9}\]3 (21) \[\sum_{j=1}^{n}z_{jik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V \tag{9}\]4 (22) \[\sum_{j=1}^{n}z_{jik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V \tag{9}\]5 (23) \[\sum_{j=1}^{n}z_{jik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V \tag{9}\]6 (24) \[\sum_{j=1}^{n}z_{jik}=\sum_{j=1}^{n}z_{ijk}\quad\forall i\in N,k\in V \tag{9}\]7  
Constraints (6)-(8) ensure that all stations \(i\in N\) must be visited exactly by a single management vehicle. Constraints (9)-(16) set up a path that begins at dummy node 0, travels to all nodes, and returns to dummy node \(n+1\). Constraints (17) and (18) eliminate sub-tours. Constraints (19) and (20) limit the number of bikes transferred by the capacity of the vehicle, denoted as \(C\), when a path is connected to two stations between \(i\) and \(j\).