Searching Strategies with Low Computational Costs for Multiple-Vehicle Bike Sharing System Routing Problem

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.