A Heteroscedastic Robust Bayesian Optimization Method for Solving Simulation-Based Transportation Problems

This study focuses on simulation-based optimization (SBO) in transportation systems considering the pervasive and influential heteroscedastic noise. Existing studies rarely consider the effects of such heteroscedasticity on the solution robustness, giving rise to suboptimal solutions that could compromise the reliability and resilience of the system in real-world applications. To address this concern, a simulation-based robust optimization problem is investigated in this study, which focuses on minimizing the expectation of simulation outputs while maintaining the stochasticity of transportation systems within predefined limits. To solve the problem and identify a robust solution under varying levels of stochasticity, a heteroscedastic robust Bayesian optimization (HRBO) method is proposed by fusing key SBO concepts and techniques with the widely used Bayesian optimization (BO) algorithm. The formulation of surrogate models, strategies for sampling new points, and evaluation issues of samples are systematically designed. Specifically, surrogate models for the stochastic objective and constraint functions are separately formulated using the Gaussian process (GP) model. To accommodate simulation noise, Bayesian posterior inference is employed to estimate objective function values and constraint function values, which are incorporated into the GP models. To locate promising feasible solutions, a constrained expected improvement (EI) function is constructed and optimized using a tailored two-stage method, which can effectively tackle the inherent issue of “flat” areas of EI functions. Considering the usually high computational cost of simulators, an adaptive simulation resource allocation scheme is designed by incorporating ranking and selection techniques into the BO framework to efficiently allocate computational resources. The proposed methods are validated on a test function and two representative simulation-based transportation problems: a variant of the M/M/1 queueing problem and a continuous network design problem. Experimental results demonstrate the superior performance of HRBO in addressing heteroscedastic noise and identifying robust solutions. Funding: This work was supported by the National Natural Science Foundation of China [Grants 52131203 and 72471057], the Jiangsu Provincial Scientific Research Center of Applied Mathematics [Grant BK20233002], and the Natural Science Foundation of Jiangsu Province [Grant BK20232019]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0840 .