In the field of neuroscience, it is widely acknowledged that neurons exhibit periodic, quasi-periodic, and chaotic responses to periodic inputs. In this study, we evaluated the responses of the Izhikevich neuron model stimulated by sinusoidal inputs. First, we analyzed the dynamical behavior of the Izhikevich neuron model to the sinusoidal inputs in the state space and found two types of responses: periodic and non-periodic. Next, we obtained the domains of the periodic and non-periodic responses on the frequency-amplitude plane of the sinusoidal inputs by evaluating the diversity index of the inter-spike intervals. Finally, we analyzed the responses of the Izhikevich neuron model using the stroboscopic plot. Consequently, we clarified that a periodic response is a limit cycle and an irregular response is a torus, which implies that the irregular responses of the Izhikevich neuron model stimulated by sinusoidal inputs are quasi-periodic responses.

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 multiplevehicles 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.

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.

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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.