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

Abstract

This study focuses on simulation-based optimization (SBO) in transportation systems considering the pervasive and influential heteroscedastic noises. 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. Existing studies predominately used surrogate-based optimization to solve such problems. However, insufficient attention has been given to the heteroscedasticity as well as the requirements for solution robustness in these methods. 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 a widely used surrogate-based optimization framework, the Bayesian optimization algorithm. The formulation of surrogate models, strategies for sampling new points, and evaluation issues of samples are systematically designed considering the effects of heteroscedastic noises. Specifically, surrogate models for the stochastic objective and constraint functions are separately formulated using the Gaussian process (GP) model. To accommodate noises of simulation, Bayesian posterior inference is employed to estimate objective function values and constraint function values, which is incorporated into the GP models. To locate and sample robust solutions, a constrained expected improvement (EI) function is constructed and optimized using a tailored two-stage method. The two-stage method, for the first time, effectively tackles 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 Bayesian optimization framework, to efficiently allocate computational resources. An allocation rule is derived in this study. The resulting methodology is validated on a variant of the M/M/1 queueing problem and a simulation-based network design problem, both of which have demonstrated the superior performance of HRBO against benchmark methods.

Keywords: Traffic simulation, Heteroscedastic noises, Simulation-based robust optimization, Bayesian optimization