A static bike rebalancing problem with optimal user incentives is investigated. The problem is formulated as a mixed-integer nonlinear and nonconvex programming model to minimize the total cost, including the travel costs, unbalanced penalties, and incentive costs. We reformulate the mixed-integer program and develop a new outer-approximation method to obtain its global $\epsilon$-optimal solutions. We also propose a bi-level variable neighborhood search algorithm to solve large problems. The results tested on small examples reveal problem properties and the performance of the outer-approximation method. The results tested on large examples show that the bi-level algorithm can provide high-quality solutions with short computational times.

研究了具有最佳用户激励的静态自行车再平衡问题。该问题被公式化为混合整数的非线性和非凸规划模型，以最小化总成本，包括旅行成本，不平衡的罚款和激励成本。我们重新制定了混合整数程序，并开发了一种新的外部逼近方法以获得其全局$\epsilon$-最佳解决方案。我们还提出了一种双层变量邻域搜索算法来解决大问题。在小例子上测试的结果揭示了问题的性质和外逼近方法的性能。在大型示例上测试的结果表明，该双层算法可以在较短的计算时间内提供高质量的解决方案。