Subsidy Allocation Problem with Bus Frequency Setting Game: A Trilevel Formulation and Exact Algorithm
研究政府如何在考虑公交运营商竞争和乘客路线选择的情况下分配补贴以最小化社会成本，建立三层优化模型并用精确算法求解，发现竞争线路通常不需补贴，且竞争行为在高时间价值城市中更有效降低社会成本。
Typically, governments subcontract the operation of urban bus systems to several bus operators. In particular, the government aims to promote the service quality for passengers by introducing competition among bus operators and subsidizes bus operations to ensure affordable fares. However, most existing studies about subsidy allocation typically do not account for the competitive factors among bus operators and thus may underestimate the associated benefits. In this study, we investigate how the government allocates subsidies to minimize social costs, taking into account the competition among bus operators and passenger route decisions. We describe this problem as a trilevel optimization model and use a game-theoretic approach to characterize the market equilibrium of bus operators. Next, we transform the trilevel model into a mixed-integer programming problem with quadratic constraints and solve it using an exact algorithm with acceleration techniques. The results of numerical experiments demonstrate the computational efficiency of the proposed algorithm. Several valuable insights are derived: First, lines served by competing bus operators typically do not require subsidies. Second, competitive behavior decreases social costs (including bus operating costs and passenger travel costs) more effectively in cities in which the passengers assign higher value to time. Third, the competitive behavior may be guided by exogenous parameters, such as ticket prices, to approximate the optimum of urban bus systems. Funding: This work was supported by the Key Project [Grant 52131203], Youth Program [Grant 72301065], and Project of International Cooperation and Exchanges [Grant 72361137006] of the National Natural Science Foundation of China.