We have already proposed a multiple-vehicle bike sharing system routing problem (mBSSRP) to adjust the number of bicycles at each port using multiple vehicles in short time. However, there are many strict constraints in the mBSSRP, thus it is difficult to obtain feasible solutions of the mBSSRP for some instances. To obtain feasible solutions of the mBSSRP, we have proposed a mBSSRP with soft constraints (mBSSRP-S) that removes some constraints from mBSSRP and appends violations to an objective function as penalties, and a searching strategy that explores both the feasible and infeasible solution spaces. Numerical experiments indicated that solving mBSSRP-S to obtain feasible solutions of mBSSRP results in better performance than solving mBSSRP directly. However, mBSSRP-S includes infeasible solutions of mBSSRP, thus the neighborhood solutions and computational costs increase. In this study, we propose search strategies with low computational costs while maintaining performance. In particular, we propose two search strategies: the first one is to reduce neighborhood solutions to obtain a feasible solution in a short time before finding a feasible solution of the mBSSRP, and the second one is to change the problem to be solved (mBSSRP or mBSSRP-S) after a feasible solution is obtained and to search good near-optimal solutions in a short time. As the first search strategy, we propose two search methods for reducing the number of neighborhood solutions in the Or-opt and the CROSS-exchange and compare their performance with our previous results. From numerical experiments, we confirmed that a feasible solution can be obtained within a short time by exploring only the normal order insertion of the Or-opt and the normal order exchange of the CROSS-exchange as the neighborhood solutions. Next, as the second search strategy after a feasible solution of mBSSRP is obtained, we propose four search methods and compare their performance with our previous results. Numerical experiments show that the search method that only searches for the normal order insertion of the Or-opt and the normal order exchange of the CROSS-exchange with hard constraints after obtaining a feasible solution can obtain short tours within a short time.

In bicycle sharing systems, many vehicles restore bicycles to ports. To construct the shortest tour of these vehicles, in a previous work, we formulated the multiple-vehicle bike sharing system routing problem (mBSSRP) and demonstrated that an optimal solution can be obtained for small-sized instances through a general-purpose mixed-integer linear programming solver. However, for large-sized instances, the optimal solution could not be found in a reasonable time frame. Therefore, to find near-optimal solutions for the mBSSRPs in a short time, in this study, we develop a method with a searching strategy, which explores both the feasible and infeasible solution spaces. To investigate the performance of the proposed method, we solve benchmark problems of mBSSRP. In addition, we compare the proposed method with the method exploring only the feasible solution space, in terms of performance. The results of the numerical experiments demonstrate that the proposed method can reach optimal solutions for almost all small-sized mBSSRP instances and that searching both the feasible and infeasible solution spaces yields good feasible solutions both for small-sized and large-sized instances.

To find the shortest tour of a transporting vehicle in a bike sharing system (BSS), a bike sharing system routing problem (BSSRP) has been proposed. In the BSSRP, a single vehicle restores the number of bicycles in stations. However, in a real system, the restoration of the number of the bicycles in each station is carried out by multiple vehicles. To decide the shortest tour of the multiple vehicles, we proposed a mathematical optimization model called the multiple-vehicles bike sharing system routing problem (mBSSRP). We present a construction method and local search methods to solve the large-size mBSSRP. The result of numerical experiments shows that the proposed method can find good solutions in a short time.

Multi-objective Optimization Problems (MOPs) attempt to find solutions simultaneously by minimizing or maximizing multiple objective functions in a trade-off relationship. As a solution method for MOPs, Cooperative Multi-Objective Differential Evolution (CMODE) has been proposed. Although the CMODE shows good performance for the MOPs, this method sometimes found partially biased solutions in a single trial. To solve this problem, we have already proposed an improved CMODE method. In this study, we comprehensively evaluate the performance of the improved method on low-dimensional and high-dimensional benchmark problems using various evaluation measures. Numerical results show that the improved method exhibits a small inverted generational distance for all low-dimensional and some high-dimensional benchmark problems, as compared to conventional solution methods.

Finding the shortest paths for packets in communication networks is one of the vital engineering challenges. Recently, a routing strategy using gravitational centrality has been proposed to find efficient transmitting routes for packets to their destinations. This routing strategy successfully mitigates the congestion of packets in the networks and effectively transmits many packets to their destinations by using gravitational centrality. However, this routing strategy is difficult to transmit the packets to their destinations if large volumes of packets flow in the communication networks because fixed costs are used to determine the shortest paths for every source-destination pair. To overcome these problems, we have already proposed a routing strategy using congestion signals. This routing strategy further mitigates the congestion of packets compared to conventional routing strategies such as the routing strategy with gravitational centrality. In this study, we comprehensively evaluate the routing strategy with the congestion signals if different types of centralities are incorporated into the costs of links and various topologies are applied to the communication network models. Numerical experiments illustrate that our routing strategy using the congestion signals mitigates the congestion of packets even if other types of centralities are incorporated. In addition, our proposed routing strategy shows effective transmitting performance for the various topologies of networks.