## 2 Problem Formulation
### Multiple-Vehicle Bike Sharing System Routing Problem with Soft Constraints  
In the mBSSRP, there are three strict constraints, namely the time limit, the capacity and the loading and supplying constraints. As for the time limit constraint, in real life, if a work can be completed through overtime within a few minutes, we would work beyond the time limit. As for the loading and supplying constraints, if we have only four bicycles and are required to supply five bicycles, in real life, we would only supply the four bicycles. In other words, the mBSSRP constraints should not be treated strictly in real-life situations. We propose a problem that treats some of the mBSSRP constraints as penalties. In this problem, violations of these constraints are added to the mBSSRP objective function (Equation (1)). The resulting problem is called mBSSRP-S. In the mBSSRP-S, the time limit constraint (Equation (10)), the loading and capacity constraints (Equation (8)) and the supplying constraint (Equation (9)) are removed. Instead, the violations of these constraints are added as penalties to obtain the following objective function for the mBSSRP-S.  
\[\text{minimize}\qquad\qquad\qquad\sum_{i\in V}\sum_{j\in V}\sum_{k\in K}t_{ij}x _{ijk}+\alpha p_{\text{T}}+\beta(p_{b+}+p_{b-}). \tag{18}\]  
In Equation (18), the first term is the sum of the travel times of all vehicles, which is the objective function of mBSSRP. The working time is the sum of the traveling times of the vehicles and the loading and supply times of the bicycles. In Equation (18), \(\alpha\) and \(\beta\) are the weights of the penalties for the time limit constraint, and the loading and supplying constraints, respectively; \(p_{T}\) is the excess working time extending beyond \(T\); \(p_{b+}\) refers to the loading and the capacity violation of the bicycles, representing the total number of bicycles that could not be loaded onto the vehicles at the pickup ports; \(p_{b-}\) refers to the supplying violation of the bicycles, representing the total number of bicycles that could not be unloaded bicycles to the delivery ports. Figure 1 shows examples of the violation of the loading and supplying bicycles. In Figure 1, the capacity of the vehicle is five. In Figure 1a,b, the vehicle must load three bicycles at port \(i\) (colored in red). In Figure 1a, the vehicle can take all bicycles because the number of bicycles loaded onto the vehicle before visiting port \(i\) is two. On the other hand, in Figure 1b, the number of bicycles loaded onto the vehicle is four. Then, the vehicle can load only one bicycle. Therefore, the two bicycles cannot be loaded onto the vehicle. The number of unloaded bicycles is two, and the loading penalty is added to the objective function, namely \(p_{b+}=2\). Figure 1c shows an example of the supplying violation. In Figure 1c, the vehicle must supply three bicycles to port \(i\) (colored in blue). However, the number of bicycles loaded onto the vehicle is only one. Therefore, the two bicycles cannot be supplied at port \(i\). The number of unsupplied bicycles is two, and the supplying penalty is added to the objective function, namely \(p_{b-}=2\). When \(p_{\text{T}}\), \(p_{b+}\) and \(p_{b-}\) are zero, the solution is a feasible mBSSRP solution. However, if one of them is positive, the solution is an infeasible mBSSRP solution. \begin{table}
\begin{tabular}{l l} \hline \hline \(y_{ik}\) & vehicle \(k\) visits port \(i\) (\(y_{ik}=1\)) or not (\(y_{ik}=0\)) \\ \(x_{ijk}\) & vehicle \(k\) moves from port \(i\) to port \(j\) directly (\(x_{ijk}=1\)) or not (\(x_{ijk}=0\)) \\ \(z_{ijk}\) & the number of bicycles loaded onto vehicle \(k\) when vehicle \(k\) moves from port \(i\) to port \(j\) \\ \(f_{ijk}\) & the number of ports visited by vehicle \(k\) before it reaches port \(j\) from port \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The four decision variables used in the mBSSRP. In Equation (18), \(\alpha\) and \(\beta\) are the weight parameters of the penalty.