Freight transport is one of the most important sectors of the national economy. The increase in the efficiency of the logistics company is ensured by the accurate organization of supplies, as well as the timely delivery of goods to the points of the route. That is why there is a need to optimize the delivery of goods. A delivery task is a transport task of delivering oversized cargo from a distribution center to a multitude of recipients located in the area of operation of a transport company. It is a generalization of the well-known traveling salesman problem, from which it differs in the condition of the limited carrying capacity of the vehicle used and, as a consequence, the need to repeatedly return to the base to replenish the transported cargo. This article proposes to supplement the traditional formulation of the problem with the requirement to distribute customers among several simultaneously operating vehicles (TC) so that the maximum lead time is minimal. This takes into account not only the interests of the delivery contractor, but also the customers. The solution to the problem consists of two stages. On the first one, in some known way, rational ring routes for each vehicle are determined, minimizing the total mileage. Based on the results of the stage, the time for passing each route is calculated. At the second stage, the problem of reducing the maximum travel time of routes is solved by using several vehicles delivering at the same time. To do this, it is necessary to optimally distribute vehicles along individual routes. It is proposed to use the algorithm for solving the well-known problem "Packing items into containers" to solve this problem. This problem belongs to the class of NP-hard problems in the strong sense and does not have an exact solution algorithm for arbitrary input data. This paper proposes a complex solution method combining the well-known First fitted (FF) algorithm, the original First fitted with reordering (FFR) algorithm, and S. Martello's lower bounds and P. Thoth to control the optimality of the obtained solution. Test calculations have shown the effectiveness of this approach for a moderate dimension of the problem.

1. Zhou L., Baldacci R., Vigo D., Wang X. A multi-depot two-echelon vehicle routing problem with delivery options arising in the last mile distribution. European Journal of Operational Research. 2018. vol. 265. no. 2. pp. 765–778. doi: 10.1016/j.ejor.2017.08.011

2. Kampf R., Hlatka M., Savin G. Proposal for optimizing specific distribution routes by means of the specific method of operational analysis. Communications-Scientific letters of the University of Zilina. 2017. vol. 19. no. 2. pp. 133–138.

3. Parhusip S.F. Composite Algorithm Based on Clarke-Wright and Local Search for the Traveling Salesman Problem. Proceedings of the 2019 5th International Conference on Industrial and Business Engineering. 2019. pp. 87–90.

4. Ivanova A.A., Chernomorova T.S. On the algorithm for solving the delivery problem and its implementation // Bulletin of Science and Practice. 2017 No. 4. P. 107–114. (in Russian).

5. Dotsenko S.I. The problem of distribution of costs during transportation along a circular route as a cooperative game. Informatics. 2017. no. 1. pp. 12–19. (in Russian).

6. Prosov S.N., Kuzmenko E.A. Decomposition of the routing problem according to the heuristics of the Clark-Wright method. World of transport. 2018. vol. 16. no. 3. pp. 190–199. (in Russian).

7. Bugaev Yu.V., Korobova L.A., Zelenova E.E. Freight traffic planning by the modified Clark-Wright method. In the collection: Virtual modeling, prototyping and industrial design. Materials of the III International Scientific and Practical Conference: Electronic resource. General edition: V.A. Nemtinov. 2016. pp. 179–182. (in Russian).

8. Bugaev Yu.V., Korobova L.A., Gudkov S.V. Solution of the problem of delivery of oversized cargo by several vehicles. Mathematical methods in engineering and technology: collection of articles. tr. international scientific. SPb, Publishing house of Polytechnic. University, 2021. pp. 55–60. (in Russian).

9. Konovalova T.V., Suprun O.S. To the question of choosing a route optimization criterion for the delivery of goods by road. Electronic network polythematic journal "Scientific works of KubSTU". 2017. no. 11. pp. 143–150. (in Russian).

10. Podshivalova K.S., Podshivalov S.F. Research of the branch and bound method in solving the problem of delivering goods along a circular route. Problems of quality and operation of vehicles. 2017. pp. 276–281. (in Russian).

11. Christensen H.I. et al. Approximation and online algorithms for multidimensional bin packing: A survey. Computer Science Review. 2017. vol. 24. pp. 63–79. doi: 10.1016/j.cosrev.2016.12.001

12. Delorme M., Iori M., Martello S. Bin packing and cutting stock problems: Mathematical models and exact algorithms. European Journal of Operational Research. 2016. vol. 255. no. 1. pp. 1–20. doi: 10.1016/j.ejor.2016.04.030

13. Boyar J., Larsen K.S., Kamali S., L?pez-Ortiz A. Online bin packing with advice. Algorithmica. 2016. no. 1. pp. 507–527. doi: 10.1007/s00453-014-9955-8

14. D?sa G., Epstein L. The tight asymptotic approximation ratio of first fit for bin packing with cardinality constraints. Journal of Computer and System Sciences. 2018. vol. 96. pp. 33–49. doi: 10.1016/j.jcss.2018.03.004

15. Asta S., ?zcan E., Parkes A.J. CHAMP: Creating heuristics via many parameters for online bin packing. Expert Systems with Applications. 2016. vol. 63. pp. 208–221. doi: 10.1016/j.eswa.2016.07.005

16. Chepikova A.V., Korobova L.A., Bugaev Yu.V. The problem of three-dimensional packing of elements. Alley of Science. 2018. vol. 1. no. 9 (25). pp. 433–436. Available at: https://www.elibrary.ru/download/elibrary_36430946_56294686.pdf (in Russian).

17. Furems E.M. Inverse problem of packing into containers in the presence of qualitative criteria – formulation and review of applied methods. Artificial Intelligence and Decision Making. 2016. no. 3. pp. 31–43. (in Russian).

18. Orlov B.Yu. Construction of an algorithm for the sequence of permutations in the research and operation of equipment of oil-extracting enterprises. Scientific research and modern education. 2017. pp. 183–185. (in Russian).

19. Shablya Yu.V., Kruchinin D.V., Repkin A.S. Comparison of approaches to the development of algorithms for combinatorial generation on the example of sets of permutations and combinations. Applied mathematics and informatics: modern research in the field of natural and technical sciences. 2019. pp. 75–80. (in Russian).

20. Titenko E.A., Kripachev A.V., Marukhlenko A.L. Switching scheme of parallel paired permutations for a specialized production device. Bulletin of the Southern Federal University. Technical science. 2018. no. 8 (202). (in Russian).