Hybrid metaheuristic optimization for multi-criteria truck–drone collaborative routing in sustainable supply chains

BackgroundLast-mile logistics accounts for approximately 28% of total logistics costs and is a major contributor to urban carbon emissions. The integration of drones with electric trucks offers a path toward more efficient and sustainable delivery, but the resulting Truck–Drone Collaborative Routing Problem (TDCRP) is NP-hard, combining binary customer-to-mode assignment, route sequencing, and drone sortie scheduling into a single combinatorial challenge that exact methods cannot solve at realistic instance sizes.ObjectiveThis paper proposes a novel Hybrid ACO+BOA algorithm—the first to couple Ant Colony Optimisation (ACO) with the Butterfly Optimisation Algorithm (BOA) for any routing problem—to simultaneously minimize truck distance, delivery makespan, and carbon emissions in the TDCRP.MethodsACO constructs 20 route sets per iteration using pheromone-heuristic guided selection. BOA refines the entire population using fragrance-scaled guided Or-opt relocation—a discrete analog of the continuous BOA position update—with greedy acceptance. Pheromone deposits are drawn from BOA-refined solutions, creating a reinforcing feedback loop. Five benchmark CVRP instances were evaluated against Pure ACO, BOA, PSO, and ABC over 20 independent runs each. A 33-configuration sensitivity analysis identified the optimal parameter set. Full non-parametric statistical validation was performed (Friedman, Nemenyi, Wilcoxon with Cohen's d).ResultsThe Hybrid achieved the lowest or statistically equivalent normalized combined objective on all five instances, with significant superiority on P-series instances. On P-n101-k4, it recorded a mean normalized objective of 0.2267 ± 0.0046—reductions of 5.2%, 63.6%, 72.4%, and 69.3% relative to ACO, BOA, PSO, and ABC, respectively. On A-series instances, the Hybrid was statistically equivalent to Pure ACO (p>0.19), reflecting structural characteristics where short per-vehicle routes leave limited scope for BOA refinement. The Hybrid was significantly superior to BOA, PSO, and ABC on all instances (p < 0.001, |d| = 13.7–47.8). Drone sorties were 1.9–3.1 × higher than non-ACO baselines, directly reducing fleet-level emission.ConclusionTightly coupling pheromone-guided construction with fragrance-scaled exploitation yields consistent, statistically validated improvements over all single-algorithm baselines. The algorithm provides a practical, scalable framework for sustainable last-mile logistics planning with computation times of 30–49 seconds remaining within daily scheduling horizons.