Towards an Optimal Bus Frequency Scheduling: When the Waiting Time Matters

Songsong Mo, Zhifeng Bao, Baihua Zheng, Zhiyong Peng

Abstract: Reorganizing bus frequencies to cater for the actual travel demands can significantly save the cost of the public transport system. Many, if not all, previous studies formulate this as a bus frequency optimization problem that tries to minimize passengers’ average waiting time. On the other hand, many investigations have confirmed that the user satisfaction drops faster as the waiting time increases. Consequently, this paper studies the bus frequency optimization problem considering the user satisfaction. Specifically, for the first time to our best knowledge, we study how to schedule the buses such that the total number of passengers who could receive their bus services within the waiting time threshold can be maximized. We propose two variants of the problem, FAST and FASTCO, to cater for different application needs and prove that both are NP-hard. To solve FAST effectively and efficiently, we first present an index-based <inline-formula><tex-math notation="LaTeX">$(1-1/e)$</tex-math></inline-formula> -approximation algorithm. By exploiting the locality property of routes in a bus network, we further propose a partition-based greedy method that achieves a <inline-formula><tex-math notation="LaTeX">$(1-\rho)(1-1/e)$</tex-math></inline-formula> approximation ratio. Then we propose a progressive partition-based greedy method to further boost the efficiency while achieving a <inline-formula><tex-math notation="LaTeX">$(1-\rho)(1-1/e-\varepsilon)$</tex-math></inline-formula> approximation ratio. For the FASTCO problem, two greedy-based heuristic methods are proposed. Experiments on a real city-wide bus dataset in Singapore have been conducted to verify the efficiency, effectiveness, and scalability of our methods in addressing FAST and FASTCO respectively.