## 3 Heuristic Methods
### Dynamically Changing Weight of Penalties  
When the penalty values in the mBSSRP-S objective function are zero, the solution is a feasible mBSSRP solution. In contrast, if one of the penalty values is larger than zero, the solution is an infeasible mBSSRP solution. To obtain feasible mBSSRP solutions from mBSSRP-S solutions, the parameters \(\alpha\) and \(\beta\) in the mBSSRP-S objective function must be set to appropriate values. If the parameters are small, the penalty becomes weak even when the solution has several constraint violations. Then, the method tends to search the infeasible solution space of mBSSRP. Thus, it is difficult to obtain a feasible mBSSRP solution. If the parameters \(\alpha\) and \(\beta\) are large, the current solution moves to a solution space that has small constraint violations, because these slight constraint violations result in a strong penalty. To perform an effective solution search for a good mBSSRP solution, the values of the parameters \(\alpha\) and \(\beta\) must be adjusted in accordance with the solution states. Figure 5: Five local search methods (**a-e**) used in this study. Black rectangles represent depots and gray and orange circles indicate ports. Letters in the circles indicate the port number. In these figures, dotted lines represent edges removed by the local search method, and green lines represent newly added edges. To adjust these values, we introduced a strategy based on a method for solving the vehicle routing problem with time window (VRPTW) [31]. The objective of the VRPTW is to find the shortest tour that serves all customers while satisfying the capacity, time limit and time window constraints associated with each customer. It is difficult to find a feasible solution for the VRPTW due to multiple constraints. To relax the constraints of the VRPTW, the vehicle routing problem with soft time windows (VRPTW) was proposed. In the VRPTSW, the time windows can be violated if a penalty is paid. To find feasible VRPTW solutions, the weight penalty for the time windows is dynamically changed to obtain many feasible VRPTW solutions [31]. Thus, for the mBSSRP-S study, we extended this method [31] to dynamically adjust the weights of the penalties. In VRPTSW [31], there is a single weight parameter, while in mBSSRP-S, there are two weight parameters \(\alpha\) and \(\beta\). Thus, we had to consider how to change the values of two parameters. To adjust the weights of the penalties dynamically, the objective function of the mBSSRP-S was modified as follows:  
\[z(t)=\sum_{i\in V}\sum_{j\in V}\sum_{k\in K}t_{ij}x_{ijk}+\alpha(t)p_{\text{T}} (t)+\beta(t)(p_{\text{q+}}(t)+p_{\text{q-}}(t)), \tag{19}\]  
where \(z(t)\) is the objective function of the mBSSRP-S at time \(t\); \(\alpha(t)\) and \(\beta(t)\) are the weights of the penalties for the time limit constraint and the loading and supplying constraints, respectively; \(p_{\text{T}}(t)\) is the excess work time at time \(t\), which extends beyond the time limit \(T\); \(p_{\text{q+}}(t)\) refers to the loading violation of the bicycles at time \(t\), where loading violation is defined as the total number of bicycles exceeding the loading capacity of the vehicle; \(p_{\text{q-}}(t)\) refers to the supplying violation, representing the total number of bicycles that could not be supplied. In this method, if a feasible solution is obtained at time \(t\), the weight parameter is decreased, making it is easy to move to an infeasible solution at time \(t+1\). However, if an infeasible solution is obtained at time \(t\), the weight parameter is increased, making it easy to move to the solution with small constraint violations at time \(t+1\). In this study, we propose two methods for adjusting the weights of the penalties. The first method (Method 1) updates the weights through the following equations:  
\[\alpha(t+1) = \left\{ \begin{array}{ll}\lambda\alpha(t),&\text{if }\ p_{\text{T}}(t)>0,\\ \mu\alpha(t),&\text{if }\ p_{\text{T}}(t)=0,\end{array} \right. \tag{20}\]  
and  
\[\beta(t+1) = \left\{