Received 30.06.2025; Revised 01.08.2025; Accepted 02.09.2025

Published 29.09.2025; First Online 30.09.2025

https://doi.org/10.34229/2707-451X.25.3.3

Previous  |  FULL TEXT (in Ukrainian)  |  Next

 

MSC 90B20, 90C27, 90C59, 68T20

Analysis of Swarm Intelligence Algorithms Used for Solving Vehicle Routing Problems

Maksym Yeher

Uzhhorod National University, Ukraine

Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.

 

Introduction. The Vehicle Routing Problem (VRP), first formulated by Danzig and Ramseur in 1959, has remained one of the most popular research subjects to date. This popularity stems from numerous factors, including its wide applicability across various economic sectors. VRP belongs to the class of NP-hard problems, implying high computational complexity in finding optimal solutions, especially for large-scale variations. Over the past 25 years, approaches to its classification and solution have evolved significantly, driven by real-world requirements and constraints, as well as advancements in optimization methods and computational power.

This article analyzes research findings from studies focused on VRP, confirming a substantial shift in researchers' attention towards metaheuristic approaches. It examines application of the most popular swarm intelligence algorithms and their variations, including hybrids, for solving VRP, and what makes them successful. Furthermore, the study investigates the correlation between sets of algorithm parameters.

The purpose of the paper is to investigate usage of swarm intelligence algorithms for solving the Vehicle Routing Problem. Paper attempts to determine what makes them effective for solving VRPs (if such) and how this is related to their parameter set. In addition, the study explores whether there is a correlation between the parameter sets of SI algorithms considered effective for VRPs.

Results. An analysis of the results of research articles on VRP was conducted, which made it possible to identify the most popular variations of VRP and rank the methods for solving them. A comprehensive analysis of the most popular SI algorithms, including their variations and hybrids, for solving VRP was conducted. Their strengths and weaknesses were analyzed, and algorithmic features that make them effective in solving VRP were identified. A correlation analysis was conducted between the optimal parameter sets of algorithms and a strong dependence of the optimal parameter sets on the specific variation of VRP being solved was revealed.

Conclusions. The analysis of the literature reveals that the Capacitated Vehicle Routing Problem (CVRP) remains the most prevalent VRP variant among researchers. Another popular variant is the Vehicle Routing Problem with Time Windows (VRPTW). Overall, there is an increasing trend in the popularity of VRP variants that incorporate real-world assumptions: Open VRP (OVRP), Dynamic VRP (DVRP), and Time-Dependent VRP (TDVRP). Often, real-life parameters such as cash transportation, small parcel delivery, waste collection, or social legislation regarding drivers' working hours, prompt researchers to develop narrow mathematical models. Unfortunately, these models are typically hard-wired to a specific problem, and some are even specifically adapted to particular test instances. The most popular methods studied in the literature are metaheuristic methods, classical heuristic methods, and exact methods.

Various Swarm Intelligence (SI) algorithms were analyzed. Their shared properties of exploration, exploitation, and resistance to local optima make them well-suited for the complex combinatorial nature of VRP. However, the choice of algorithm and its parameters is strongly interrelated with the specific VRP variation, emphasizing the need for integrated approaches to their selection and tuning. Despite significant progress, challenges remain in effectively solving large-scale real-world VRPs.

 

Keywords: Vehicle Routing Problem, swarm intelligence, metaheuristic methods, logistics.

 

Cite as: Yeher M. Analysis of Swarm Intelligence Algorithms Used for Solving Vehicle Routing Problems. Cybernetics and Computer Technologies. 2025. 3. P. 37–52. (in Ukrainian) https://doi.org/10.34229/2707-451X.25.3.3

 

References

           1.     Hao D., Mao X., Wei Y., Sun L., Yang Y. Review of research on vehicle routing problems. International Conference on Smart Transportation and City Engineering (STCE 2023). 2024. 13018. https://doi.org/10.1117/12.3024185

           2.     Hulianytskyi L.F., Kotkova A. A To the classification of vehicles routing problems. Scientific bulletin of Uzhhorod National Univericty. “Mathematics & Informatics” Series, 1 (36), P. 73–84. (in Ukrainian) https://doi.org/10.24144/2616-7700.2020.1(36).73-84

           3.     Braekers K., Ramaekers K., Nieuwenhuyse I. The Vehicle Routing Problem: State of the Art Classification and Review. Computers & Industrial Engineering. 2016. 99. P. 300–313. https://doi.org/10.1016/j.cie.2015.12.007

           4.     Tan S., Yeh W. The Vehicle Routing Problem: State-of-the-Art Classification and Review. Applied Sciences. 2021. 11 (21). 10295. https://doi.org/10.3390/app112110295

           5.     Yeher M. Results of critical review of the papers on solving vehicle routing problem. Mat to XXVII Internation practical seminar “Combinatorial configurations and their application”. 2025. P. 20–26. (in Ukrainian)

           6.     Kumar S., Panneerselvam R. A Survey on the Vehicle Routing Problem and Its Variants. Intelligent Information Management. 2012. 4. P. 66–74. https://doi.org/10.4236/iim.2012.43010

           7.     Yang J., Qu L., Shen Y., Shi Y., Cheng S., Zhao J., Shen X. Swarm Intelligence in Data Science: Applications, Opportunities and Challenges. Advances in Swarm Intelligence. 2020. 12145. P. 3–14. https://doi.org/10.1007/978-3-030-53956-6_1

           8.     Abdelaziz D. The Impact of Swarm Intelligence on Optimization Problems. Int J Swarm Evol Comput. 2024. 13 (5). P. 390. https://doi.org/10.35248/2090-4908.24.13.390

           9.     Abadlia H., Belhassen I., Smairi N. Adaptive multi-strategy particle swarm optimization for solving NP-hard optimization problems. International Journal of Knowledge-Based and Intelligent Engineering Systems. 2024. 28 (1). P. 195–209. https://doi.org/10.3233/KES-230137

       10.     Shi Y., Meng F., Shen G. A Modified Artificial Bee Colony Algorithm for Vehicle Routing Problems with Time Windows. Information Technology Journal. 2012. 11. P. 1490–1495. https://doi.org/10.3923/itj.2012.1490.1495

       11.     Yang X., Xingshi H. Firefly Algorithm: Recent Advances and Applications. International Journal of Swarm Intelligence. 2013. 1. https://doi.org/10.1504/IJSI.2013.055801

       12.     Vasuki A. Nature-Inspired Optimization Algorithms (1st ed.). Chapman and Hall/CRC. 2020. P. 166–180. https://doi.org/10.1201/9780429289071

       13.     Wu H., Zhang F., Wu L. New swarm intelligence algorithm-wolf pack algorithm. Systems Engineering and Electronics. 2013. 35. https://doi.org/10.3969/j.issn.1001-506X.2013.11.33

       14.     Tian A.-Q., Chu S.-C., Pan J.-S., Liang Y. A Novel Pigeon-Inspired Optimization Based MPPT Technique for PV Systems Processes. 2020. 8 (3). P. 356. https://doi.org/10.3390/pr8030356

       15.     Rizzoli A., Montemanni R., Lucibello E., Gambardella L. Ant colony optimization for real-world vehicle routing problems. Swarm Intelligence. 2006. 1. P. 135–151. https://doi.org/10.1007/s11721-007-0005-x

       16.     Hulianytskyi L., Rybalchenko O. Route Optimization in Mission Planning for Hybrid DRONE+VEHICLE Transport Systems. Cybernetics and Computer Technologies. 2023. 3. P. 44–58. (in Ukrainian) https://doi.org/10.34229/2707-451X.23.3.4

       17.     Mar’I F., Ubaidillah H., Mahmudy W. Hybrid Artificial Bee Colony Algorithm and Improved Simulated Annealing for Capacitated Vehicle Routing Problem. Knowledge Engineering and Data Science (KEDS). 2022. 5 (2). P. 109–121. https://doi.org/10.17977/UM018V5I22022P109-121

       18.     Singh G., Prateek M., Kumar S., Verma M., Singh D., Lee H. Hybrid Genetic Firefly Algorithm-based Routing Protocol for VANETs. IEEE Access. 2022. 10. P. 1. https://doi.org/10.1109/ACCESS.2022.3142811

       19.     Cut E., Xu H., Wong W. Metaheuristics is All You Need. 2025. https://arxiv.org/html/2411.05797v2 (звернення: 25.06.2025)

       20.     Mahmudy W.F., Widodo A.W., Haikal A.H. Challenges and Opportunities for Applying Meta-Heuristic Methods in Vehicle Routing Problems: A Review. Engineering Proceedings. 2024. 63. P. 12. https://doi.org/10.3390/engproc2024063012

       21.     Zhang X., Wei Y., Hashim Z. Improvement of Swarm Intelligence Algorithm and Its Application in Logistics Network Routing. Taiwan Ubiquitous Information, Journal of Network Intelligence. 2023. 8 (4). P. 1077–1094. https://bit.kuas.edu.tw/~jni/2023/vol8/s4/02.JNI-S-2023-04-014.pdf (звернення: 25.06.2025)

       22.     Yu N., Jiang W., Hu R., Qian B., Wang L. Learning Whale Optimization Algorithm for Open Vehicle Routing Problem with Loading Constraints. Discrete Dynamics in Nature and Society. 2021. P. 1–14. https://doi.org/10.1155/2021/8016356

       23.     PhamV., Nguyen V., Dang N. Hybrid whale optimization algorithm for enhanced routing of limited capacity vehicles in supply chain management. Scientific Reports. 2024. 14. https://doi.org/10.1038/s41598-024-51359-2

       24.     Mirjalili S., Lewis A. The Whale Optimization Algorithm. Advances in Engineering Software. 2016. 95. P. 51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008

       25.     Dorigo M. Ant colony optimization. Scholarpedia. 2007. 2 (3). P. 1461. http://doi.org/10.4249/scholarpedia.1461

       26.     Dorigo M., Birattari M., Stützle T. Ant Colony Optimization. Computational Intelligence Magazine, IEEE. 2006. 1. P. 28–39. https://doi.org/10.1109/MCI.2006.329691

       27.     Krentz T., Greenhagen C., Roggow A., Desmond D., Khorbotly S. A modified ant colony optimization algorithm for implementation on multi-core robots. Swarm/Human Blended Intelligence Workshop (SHBI). 2015. P. 1–6. http://doi.org/10.1109/SHBI.2015.7321683

       28.     Alinezhad H., Yaghoubi S., Hosseini-Motlagh S.-M., Allahyari S., Saghafi M. An Improved Particle Swarm Optimization for a Class of Capacitated Vehicle Routing Problems. International Journal of Transportation Engineering. 2018. 5. P. 331–347. https://www.researchgate.net/publication/321299808_An_Improved_ Particle_Swarm_Optimization_for_a_Class_of_Capacitated_Vehicle_Routing_Problems (звернення: 25.06.2025)

       29.     Wei J., Gu Y., Lu B., Cheong N. RWOA: A novel enhanced whale optimization algorithm with multi-strategy for numerical optimization and engineering design problems. PLoS ONE. 2025. 20 (4). https://doi.org/10.1371/journal.pone.0320913

       30.     Salehahmadi Z., Manafi A. How can bee colony algorithm serve medicine? World J Plast Surg. 2014. 3 (2). P. 87–92. https://pmc.ncbi.nlm.nih.gov/articles/PMC4236990 (звернення: 25.06.2025)

       31.     Johari N., Zain A., Mustaffa N., Udin A. Firefly Algorithm for Optimization Problem. Applied Mechanics and Materials. 2013. 421. P. 512–517. https://doi.org/10.4028/www.scientific.net/AMM.421.512

       32.     O M.Z., Mu`azu M.B., Adedokun A.E., Tijani S.A., Bello I.A. Optimized Model Simulation of a Capacitated Vehicle Routing problem based on Firefly Algorithm. Covenant Journals Informatics and Communication Technology. 2018. 6 (2). https://journals.covenantuniversity.edu.ng/index.php/cjict/article/view/1177 (звернення: 25.06.2025)

       33.     Wei X., Xiao Z., Wang Y. Solving the Vehicle Routing Problem with Time Windows Using Modified Rat Swarm Optimization Algorithm Based on Large Neighborhood Search. Mathematics. 2024. 12. P. 1702. https://doi.org/10.3390/math12111702

       34.     Wu H.-S., Zhang F.-M. Wolf Pack Algorithm for Unconstrained Global Optimization. Mathematical Problems in Engineering. 2014. P. 1–17. https://doi.org/10.1155/2014/465082

       35.     Geetha S., Poonthalir G., Vanathi P.T. A Hybrid Particle Swarm Optimization with Genetic Operators for Vehicle Routing Problem. Journal of Advances in Information Technology. 2010. 1 (4). P. 181–188. https://doi.org/10.4304/jait.1.4.181-188

       36.     Zhang H., Xiao L., Chao M., Liangzhong C. An Elite Wolf Pack Algorithm Based on the Probability Threshold for a Multi-UAV Cooperative Reconnaissance Mission. Drones. 2024. 8 (9). P. 513. https://doi.org/10.3390/drones8090513

       37.     Wu H., Xiao R. Flexible Wolf Pack Algorithm for Dynamic Multidimensional Knapsack Problems. Research (Wash D C). 2020. 2020. https://doi.org/10.34133/2020/1762107

       38.     Li C., Duan H. Target detection approach for UAVs via improved Pigeon-inspired Optimization and Edge Potential Function. Aerospace Science and Technology. 2014. 39. P. 352–360. https://doi.org/10.1016/j.ast.2014.10.007

       39.     Yang Z., Duan H., Fan Y., Deng Y. Automatic Carrier Landing System multilayer parameter design based on Cauchy Mutation Pigeon-Inspired Optimization. Aerospace Science and Technology. 2018. 79. https://doi.org/10.1016/j.ast.2018.06.013

       40.     Hu C., Xia Y., Zhang J. Adaptive Operator Quantum-Behaved Pigeon-Inspired Optimization Algorithm with Application to UAV Path Planning. Algorithms. 2018. 12. P. 3. https://doi.org/10.3390/a12010003

       41.     Yunpeng M., Wang X., Meng W. A Reinforced Whale Optimization Algorithm for Solving Mathematical Optimization Problems. Biomimetics. 2024. 9 (9). P. 576. https://doi.org/10.3390/biomimetics9090576

       42.     Mohammed H.M., Umar S.U., Rashid T.A. A Systematic and Meta-Analysis Survey of Whale Optimization Algorithm. Computational Intelligence and Neuroscience. 2019. 8718571. https://doi.org/10.1155/2019/8718571

       43.     Junhao W., Gu Y., Yan Y., Li Z., Lu B., Pan S., Cheong N. LSEWOA: An Enhanced Whale Optimization Algorithm with Multi-Strategy for Numerical and Engineering Design Optimization Problems. Sensors. 2025. 25 (7). P. 2054. https://doi.org/10.3390/s25072054

       44.     Jedrzejowicz P., Wierzbowska I. Parallelized Swarm Intelligence Approach for Solving TSP and JSSP Problems. Algorithms. 2020. 13 (6). P. 142. https://doi.org/10.3390/a13060142

       45.     Stamadianos T., Taxidou A., Marinaki M., Marinakis Y. Swarm intelligence and nature inspired algorithms for solving vehicle routing problems: a survey. Operational Research. 2024. 24. P. 47. https://doi.org/10.1007/s12351-024-00862-5

       46.     Shen Y., Liu M., Yang J., Shi Y. Middendorf M., A Hybrid Swarm Intelligence Algorithm for Vehicle Routing Problem With Time Windows. IEEE Access. 2020. 8. P. 93882–93893. https://doi.org/10.1109/ACCESS.2020.2984660

       47.     Alexander A., Sriwindono H. The Comparison of Genetic Algorithm and Ant Colony Optimization in Completing Travelling Salesman Problem. ICSTI EAI. 2020. https://doi.org/10.4108/eai.20-9-2019.2292121

       48.     Zixuan X., Xueyu H., Wenwen L. Subpopulation Particle Swarm Optimization with a Hybrid Mutation Strategy. Computational Intelligence and Neuroscience. 2022. 9599417. https://doi.org/10.1155/2022/9599417

       49.     Joy G., Huyck C., Yang X.-S. Parameter Tuning of the Firefly Algorithm by Three Tuning Methods: Standard Monte Carlo, Quasi-Monte Carlo and Latin Hypercube Sampling Methods. 2025. https://doi.org/10.1016/j.jocs.2025.102588

       50.     Abdolhosseinzadeh M., Alipour M.-M. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control and Optimization. 2021. 11 (2). P. 321–332. https://goi.org/10.3934/naco.2020028

       51.     Gendreau M., Potvin J.-Y., Bräysy O., Hasle G., Løkketangen A. Metaheuristics for the Vehicle Routing Problem and Its Extensions: A Categorized Bibliography. The Vehicle Routing Problem: Latest Advances and New Challenges. 2008. 43. P. 143–169. https://doi.org/10.1007/978-0-387-77778-8

 

 

ISSN 2707-451X (Online)

ISSN 2707-4501 (Print)

Previous  |  FULL TEXT (in Ukrainian)  |  Next