## 3 Heuristic Methods
### Construction Method  
We introduce two construction methods. The first method constructs a feasible solution for the mBSSRP and the second constructs a feasible solution for the mBSSRP-S. The first construction method sequentially inserts a port into a current tour until all ports are visited by a vehicle. We refer to the first construction method as \"the cheapest insertion method\". When a port is inserted into the current tour, the resulting new tour must satisfy the constraints of the mBSSRP. The algorithm of the first construction method is described as follows:  
1. [label=0.,leftmargin=*]
2. Port \(i\) is randomly selected. If the number of bicycles at port \(i\) is negative (\(b_{i}<0\)), the vehicle departs from the depot after loading \(b_{i}\) bicycles, supplies \(b_{i}\) bicycles to port \(i\) and returns to the depot. However, if \(b_{i}>0\), the empty vehicle goes to port \(i\) to load \(b_{i}\) bicycles and returns to the depot. In Figure 2(a), the red port is a randomly selected port, and the pink vehicle goes to the selected red port and returns to the depot. 3. An unvisited port \(k\) is inserted into the current tour. To insert the unvisited port into an edge \((i,j)\) in the current tour, the cost \(\Delta_{ikj}=t_{ik}+t_{kj}-t_{ij}\) is calculated for all unvisited ports \(k\). If the sub-tour after inserting port \(k\) into the edge \((i,j)\) cannot satisfy the constraints of the mBSSRP, this inserting operation is prohibited. 4. Port \(k\), where the cost \(\Delta_{ikj}\) is minimum, is inserted into edge \((i,j)\). In Figure 2(b), the red port is inserted into the tour of the pink vehicle. 5. When no ports satisfy the constraints, another vehicle goes to the unvisited port nearest to the depot. Figure 2(c) shows such a case; the red port is inserted into the tour of the pink vehicle; however, the time limit constraint is not satisfied. Therefore, a new vehicle (yellow) goes to the unvisited port nearest to the depot and returns to the depot as shown in Figure 2(d). Figure 2: Flowchart of the proposed method. We construct an initial tour using the construction method. Then, we improve the initial tour using the inserting method and the swapping method. Next, the CROSS-exchange and the Or-opt are controlled by the tabu search and aim to transition to a feasible solution of mBSSRP. After feasible solutions of mBSSRP are obtained in the tabu search process, the 2-opt is executed. If we cannot obtain a feasible solution, the 2-opt method is not executed. Finally, to obtain better solutions, we execute the inserting method and the swapping method again. 5. Steps 2 to 4 are repeated until all ports are visited once by one of the vehicles (Figure 3e). 6. If all vehicles are already used and there exist unvisited ports, the method cannot construct a feasible mBSSRP solution. In Figure 3f, the red port cannot be visited by any vehicle. This is an infeasible solution. The second construction method is used for the construction of a feasible mBSSRP-S solution. We refer to this as \"the farthest insertion method\". It is often used as one of the construction methods for solving the traveling salesman problem. The algorithm of the second construction method is described as follows:  
1. We randomly select \(m\) ports, where \(m\) is the number of vehicles. In Figure 4, the number of vehicles is three. Each vehicle goes to one of the selected ports and returns to the depot. In Figure 4a, three red ports are randomly selected. In Figure 4b, the blue, yellow and pink vehicles go to one of the selected ports and return to the depot. 2. An unvisited port \(k\), which is the farthest from the depot, is selected. In Figure 4c, the farthest port from the depot is shown in green. 3. Port \(k\) is inserted between port \(i\) and port \(j\) in the tour, where \(\Delta_{ikj}=t_{ik}+t_{kj}-t_{ij}\) is minimized, which means that an increase in the tour length after the insertion of the farthest port is minimum. In Figure 4d, the farthest port (green) is inserted into the tour of the blue vehicle. 4.