Time-of-day pricing for toll roads under traffic demand uncertainties: A distributionally robust simulation-based optimization method

Road pricing has received increasing attention for traffic congestion mitigation, but traffic demand uncertainties cause a profound impact on its effectiveness. This study proposes a link-based time-of-day pricing distributionally robust simulation-based optimization (DRSO) model to minimize the worst-case expected total travel time (TTT). This model aims to answer where and how much to charge under origin–destination (OD) traffic demand uncertainties that are described by an ambiguity set of probability distributions, and is solved by a DRSO method with two stages. The first stage searches an infill sample after building two stochastic kriging models (i.e., SKG-I and SKG-II) to map the relation between the traffic demand uncertainty parameter and the average TTT for each decision sample, and that between the decision variable and the worst-case average TTT, respectively. The second stage introduces the optimal computational budget allocation (OCBA) method to distribute more simulation replications to more promising combination samples, which helps to improve the prediction accuracy of SKGs in promising regions. The DRSO method is first tested with an M/M/1 queuing problem, and numerical results show its outperformance compared with one counterpart algorithm. After that, it is applied to address the link-based time-of-day pricing DRSO problem on Anaheim network and two solution algorithms are also added for comparison. Numerical results show that the robust toll plan solved by the DRSO method outperforms no toll plan and two other toll plans solved by the comparison algorithms in handling traffic dynamics and different levels of traffic demand uncertainties. In conclusion, the DRSO method is promising to address the costly simulation-based optimization (SO) problems under uncertainties characterized by an ambiguity set of probability distributions.