Stochastic multi-objective optimization (SMOOP) requires ranking multivariate distributions; yet, most empirical studies perform scalarization, which loses information and is unreliable. Based on the optimal transport theory, we introduce the center-outward q-dominance relation and prove it implies strong first-order stochastic dominance (FSD). Also, we develop an empirical test procedure based on q-dominance, and derive an explicit sample size threshold, n(δ), to control the Type I error. We verify the usefulness of our approach in two scenarios: (1) as a ranking method in hyperparameter tuning; (2) as a selection method in multi-objective optimization algorithms. For the former, we analyze the final stochastic Pareto sets of seven multi-objective hyperparameter tuners on the YAHPO-MO benchmark tasks with q-dominance, which allows us to compare these tuners when the expected hypervolume indicator (HVI, the most common performance metric) of the Pareto sets becomes indistinguishable. For the latter, we replace the mean value-based selection in the NSGA-II algorithm with q-dominance, which shows a superior convergence rate on noise-augmented ZDT benchmark problems. These results establish center-outward q-dominance as a principled, tractable foundation for seeking truly stochastically dominant solutions for SMOOPs.

Solving multi-objective vehicle routing problem with time windows (VRPTW) is a challenge in last-mile logistics. While traditional methods have predominantly focused on minimizing travel distances and vehicle numbers, recent innovations acknowledge the need for a more refined balance across various objectives. These advancements incorporate a range of heuristic algorithms, blending hybrid techniques that merge genetic algorithms with local search strategies, and employing advanced methodologies such as particle swarm optimization and multi-objective evolutionary algorithms based on decomposition (MOEA/D). Such developments signify a shift towards more effective solutions for VRPTW. In response to these challenges, this paper introduces a novel approach that integrates MOEA/D with K-means clustering for generating efficient solutions. In our approach, we refine the tri-objective VRPTW efficiency by establishing a criterion that prioritizes customer sequences based on their spatial proximity and angular alignment relative to cluster centers and depot, utilizing an integrated metric of distance and cosine angles, which can enhance the local search process and effectively balance the three objectives in VRPTW. Experimental results on Solomon's 56 VRPTW 100-customer dataset instances demonstrate the superiority of our approach.