Hybrid whale optimization algorithm for enhanced routing of limited capacity vehicles in supply chain management

www.nature.com/scientificreports OPEN Hybrid whale optimization algorithm for enhanced routing of limited capacity vehicles in supply chain management Vu Hong Son Pham , Van Nam Nguyen * & Nghiep Trinh Nguyen Dang The present study focuses on the problem of vehicle routing with limited capacity, with the objective of minimizing the transportation distance required to serve h clients with predetermined locations and needs. The aim is to create k trips that cover the shortest possible distance. To achieve this goal, a hybrid whale optimization algorithm (hGWOA) is proposed, which combines the whale optimization algorithm (WOA) with the grey wolf optimizer (GWO). The proposed hybrid model is comprised of two main steps. First step, the GWO’s hunting mechanism is integrated transitioning to the utilization phase of WOA, and a newly devised state is introduced that is linked to GWO. In the second step, a novel technique is incorporated into the exploration mission phase to enhance the resolve after per iteration. The algorithm’s performance is assessed and compared with other modern algorithms, including the GWO, WOA, ant lion optimizer (ALO), and dragonfly algorithm (DA) using 23 benchmark test functions and CEC2017 benchmark test function. The results indicate that the hybrid hGWOA method outperforms other algorithms in terms of delivery distance optimization for scenarios involving scale and complexity. These findings are corroborated through case studies related to cement delivery and a real-world scenario in Viet Nam. The vehicle routing problem (VRP) with limited capacity serves as a complex extension of the classic traveling salesman problem (TSP). In this context, the objective is to outline k routes, optimizing for minimal cost or distance, to cater to ℎ clients, each with their predetermined locations and demands. It’s crucial that each vehicle starts and concludes its journey at a specified point, all while adhering to particular constraints. Numerous methodologies have been proposed to tackle the VRP challenge. These include linear programming, the ant lion optimizer (ALO), particle swarm optimization (PSO), modified hybrid particle swarm optimization (MHPSO), double population genetic algorithm (DPGA), whale optimization algorithm (WOA), grey wolf optimizer (GWO), genetic algorithm (GA), and the dragonfly algorithm (DA). In the realm of transportation and logistics, the VRP stands as a paradigmatic NP-hard challenge. Despite being the subject of extensive academic investigation, characterizing the VRP remains elusive due to its intricate array of constraints and stipulations. These include factors like Chronological Span, Length, Collection and Drop-off, and Capability, as outlined by Laporte1. As a result, research endeavors addressing the VRP are tasked with focusing on pivotal parameters such as l ength2, cost3, and the intertwined factors of temporal duration and carbon emissions4. Liu et al.5 differentiated the VRP from the TSP by highlighting the former’s provision for multiple routes. Each of these routes is constrained by a specific vehicle capacity and must traverse all nodes. Given the daunting complexity inherent to the VRP, research has chiefly gravitated towards heuristic and metaheuristic strategies as the primary methodologies to derive workable solutions. The VRP has ascended as a key subject in academic research, chiefly due to its pivotal role in transportation and logistics. Given the necessity to ensure punctual deliveries of large volumes of goods, the task often exceeds the capabilities of individual vehicles. Taking into account each vehicle’s inherent capacity and load restrictions, devising astute delivery routes becomes essential to meet daily consumer demands. Researchers in this arena endeavor to calibrate the objective function, targeting an optimal solution that simultaneously minimizes costs, geographical span, time constraints, and carbon emissions. Such optimization efforts encompass a range of approaches, from tackling the VRP in contexts where goods are dispatched from a single depot6,7 to more intricate setups originating from multiple d epots8,9. Department of Construction Engineering and Management, Ho Chi Minh City University of Technology (HCMUT), Vietnam National University (VNU-HCM), Ho Chi Minh City, Vietnam. *email: Scientific Reports | (2024) 14:793 | https://doi.org/10.1038/s41598-024-51359-2 1 Vol.:(0123456789) www.nature.com/scientificreports/ The significance of optimization is evident across a myriad of fields, leading to a marked increase in the focus on metaheuristic techniques. One of the salient features of metaheuristics is their adaptability. From a broader perspective, metaheuristics can be delineated based on the degree of randomness they introduce during each optimization iteration. They can also be characterized based on their foundational inspirations, many of which are derived from swarm intelligence. Examples include the whale optimization algorithm (WOA)10, the grey wolf optimizer (GWO)11, and the African wild dog optimization algorithm (AWDO)12. These metaheuristic techniques find applications in diverse domains, such as the time–cost trade-off in construction p rojects13,14, dispatching of ready-mix concrete trucks15, optimization of construction site layouts16, VRP17, reduction of construction material costs18, logistics cost o ptimization19, and the design optimization of water distribution systems20. The WOA, a metaheuristic optimization technique, was introduced by Mirjalili and Lewis10 in 2016. Deriving its inspiration from the intricate hunting behaviors of humpback whales, this method employs a set of candidate solutions, each representing a potential optimum. The WOA unfolds through a three-pronged schema of search strategies: exploration, exploitation, and convergence. In the exploration phase, the algorithm adopts a stochastic approach, identifying promising regions within the vast search space. As it shifts to the exploitation stage, it mirrors the humpback’s bubble-net hunting tactics to close in on these pinpointed regions. Finally, in the convergence phase, the WOA concentrates on the fine-tuning of the best solution, progressively narrowing the search scope. Despite inherent limitations, such as sensitivity to parameter variations and a tendency towards premature convergence, the WOA is lauded for its versatility, user-friendly nature, and notable efficiency in diverse sectors, including engineering, electrical systems, and finance. The WOA has garnered significant attention due to its wide applicability across diverse domains. Correspondingly, there has been a surge in research initiatives aimed at refining its optimization capabilities. Chakraborty, Saha21 unveiled a modified WOA (mWOAPR) to enhance the diagnosis of COVID-19 severity using chest X-ray images. Notably, their findings outperformed both the foundational and other advanced metaheuristic algorithms, espec (...truncated)