Optimal Vehicle Sampling for Fleet-Wide Emissions Monitoring using Submodular Maximization

This article addresses the problem of optimal vehicle sampling for fleet-wide in-use emissions monitoring, a necessity driven by the absence of direct emissions sensors in modern production vehicles and the variable impact of in-use changes and operational factors (mileage, time-in-service, workload) on emissions performance across a fleet. Recognizing that comprehensive fleet testing is impractical due to significant downtime and cost, we propose a novel approach to identify a small, yet optimally informative subset of vehicles for sampling. The proposed approach leverages submodular function maximization, a technique rooted in optimal experimental design, specifically D-optimal design, to maximize the determinant of the information matrix (e.g., of $X^TX$, where $X$ is the regressor/design matrix in the case of a linear in parameters model). This approach ensures that the collected data yields maximum information for refining and building accurate models for emissions characteristic changes. We compare the submodular maximization strategy with conventional uniform and extreme sampling methods. Our simulation results demonstrate the potential for the submodular approach to outperform both alternatives by achieving lower variance (as measured by standard deviation and coefficient of variation) in estimating parameters for the assumed linear, quadratic, and simplified quadratic models for emission characteristic changes. The application of submodular function maximization is thus shown to be beneficial in vehicle fleet management for data collection in resource-constrained environments and leading to more accurate in-use emissions prediction. The envisioned process, in which a limited number of vehicles selected by our methodology are tested and the data are utilized to improve emissions models, can support the implementation of model-based strategies for engine emissions management.