Solving the logistics problem in a quantile formulation with a limitation on the delivery time

Context and relevance. The article discusses the quantile statement of the logistics problem with a limitation on the task execution time. The problems of transport logistics have long been well studied in deterministic formulation, however, taking into account the probabilistic limitation on the time of delivery of cargo, the problem translates into the class of stochastic optimization problems and requires the development of special solution methods. Objective. For the company, it is necessary to transport the same type of cargo from several warehouses to the final consumers. The delivery time for each consumer is limited. It is required to minimize the costs associated with the delivery of goods, taking into account the fact that the travel time from each warehouse to each end consumer is stochastic. A number of vehicles are assigned to each warehouse. All vehicles are of the same type and within the framework of the task they carry out only one delivery from the warehouse to one consumer. Hypothesis. To solve the problem, the development of a special solution algorithm is required, since the use of algorithms for solving deterministic optimization problems run into difficulties associated with a large dimension of an equivalent deterministic problem. Methods and materials. The problem is formulated in terms of stochastic linear programming with a quantile criterion and an optimization strategy in the form of a matrix of Boolean variables. The confidence level reflects the likelihood of meeting a joint time limit on the delivery of goods to each of the consumers. To solve the problem, an effective solution algorithm is proposed. Results. The results of the numerical experiment are given, reflecting the effectiveness of the algorithm. Conclusions. The shown results show the effectiveness of the proposed algorithm in comparison with standard deterministic optimization algorithms used to solve the equivalent deterministic problem given in the work.

1. Агапова, Е.Г., Попова, Т.М. (2021). Математическая модель задачи логистики с переменным тарифом. International Journal of Advanced Studies: Transport and Information Technologies, 11(2), 7-20. (In Russ.). DOI: 10.12731/2227-930X-2021-11-2-7-20
   Agapova, E.G., Popova, T.M. (2021). Mathematical model of the problem of logistics with a variable tariff. International Journal of Advanced Studies: Transport and Information Technologies, 11(2), 7-20. (In Russ.). https://doi.org/12731/2227-930X-2021-11-2-7-20
2. Ашманов, С.А. Линейное программирование. (2021). URSS. Изд. 2. стереотип.
   Ashmanov S.A. Linear programming. (2021). URSS. Publ. 2, Stereotyp. (In Russ.).
3. Босов, А.В., Мхитарян, Г.А., Наумов, А.В., Сапунова, А.П. (2019). Использование гамма-распределения в задаче формирования ограниченного по времени теста. Информатика и ее применение, 13(4), 12-18. (In Russ.).  https://doi.org/10.14357/19922264190402 
   Bosov, А.V.,  Naumov, A.V., Mhitarian, G.A., Sapunova А.P. (2019). Using gamma distribution in a time-limited test problem.  Informatics and its application (Russia),13(4), 12–18. (In Russ.). https://doi.org/10.14357/19922264190402
4. Гайнанов, Д.Н., Игнатов, А. Н., Наумов, А. В., Рассказова, В. А. (2020). О задаче назначения “технологического окна” на участках железнодорожной сети. Автоматика и Телемеханика, 6, 3-16. (In Russ.).  https://doi.org/10.31857/S0005231020060013
   Gainanov, D.N., Ignatov, A.N., Naumov, A.V., Rasskazova, V.A. (2020). On track procession assignment problem at the railway network sections. Automation and Remote Control, 81(6), 967-977. https://doi.org/10.1134/S0005117920060028
5. Игнатов, А.Н. (2020). О формировании позиционного управления в многошаговой задаче портфельной оптимизации с вероятностным критерием. Автоматика и телемеханика,12, 50–66. (In Russ.).  https://doi.org/10.31857/S000523102012003X
   Ignatov, A.N. (2020). On the construction of positional control in the multistep portfolio optimization problem with probabilistic criterion. Automation and Remote Control, 81(12), 2181-2193.
6. Кан, Ю.С., Кибзун, А.И. (2009). Задачи стохастического программирования с вероятностными критериями. М.: Физматлит.
   Kan, Yu.S., Kibzun, А.I. (2009). Stochastic programming problems with probabilistic criteria. М.: Fizmatlit. (In Russ.).
7. Кибзун, А.И., Наумов, А.В., Норкин, В.И. (2013). О сведении задачи квантильной оптимизации с дискретным распределением к задаче смешанного целочисленного программирования. Автоматика и Телемеханика, 6, 66–86. (In Russ.). https://doi.org/10.1134/S0005117913060064.
   Kibzun, А.I., Naumov, А.V., Norkin, V.I. (2013). Оn reducing a quantile optimization problem with discrete distribution to a mixed integer programming problem. Automation and Remote Control, 74(6), 951-967.   https://doi.org/10.1134/S0005117913060064.
8. Наумов, А.В., Мартюшова, Я. Г., Степанов, А. Е. (2024). Оптимизация прохождения ограниченного по времени теста по квантильному критерию. Информатика и её применения, 18(4), 44–51. (In Russ.).  https://doi.org/10.14357/19922264240406
   Naumov, A.V.,  Martyushova, Ya.G., Stepanov, A.E. (2024).  Optimization of passing a time-limited test according to the quantile criterion.  Informatics and its application (Russia),18(4), 44–51. (In Russ.).  https://doi.org/10.14357/19922264240406
9. Наумов, А.В., Богданов, А.Б. (2006). Решение двухэтапной задачи логистики в квантильной постановке. Автоматика и Телемеханика, 12, 36-42. (In Russ.). 
   Naumov, A.V., Bogdanov, A.B. (2006). Solution to a two-step logistic problem in a quantile statement. Automation and Remote Control, 67(12),1893-1899.   https://doi.org/10.1134/S0005117906120034
10. Наумов, А.В., Игнатов, А.Н. (2022). Решение задач стохастического линейного программирования с квантильным критерием. М: Доброе слово и Ко.
    Naumov, A.V., Ignatov, А.N. (2022). Solving stochastic linear programming problems with quantile criterion.  М: Kind word and Co. (In Russ.).
11. Наумов, А.В., Мхитарян, Г.А., Черыгова, Е.Е. (2019). Стохастическая постановка задачи формирования теста заданного уровня сложности с минимизацией квантили времени выполнения. Вестник компьютерных и информационных технологий, 2, 37–46. DOI: 14489/vkit.2019.02.pp.037-046.
    Naumov, A.V., Mhitarian, G.A., Cherygova, E.E. (2019). Stochastic formulation of the task of forming a test of a given complexity level with minimization of quantile of execution time. Bulletin of Computer and Information Technologies (Russia), 2. 37–46. (In Russ.). DOI: 10.14489/vkit.2019.02.pp.037-046.
12. Наумов, А.В., Уланов, С.В. (2003). Учет риска в двухэтапных задачах оптимального распределения ресурсов. Автоматика и Телемеханика, 7, 109-116. (In Russ.).
    Naumov, A.V., Ulanov, S.V. (2003). Risk in two-stage  optimal resource allocation. Automation and Remote Control, 64(7),1115-1121.  https://doi.org/10.1023/A:1024786218814
13. Наумов, А.В., Устинов, А. Э., Степанов, А.Е. (2024). О задаче максимизации вероятности успешного прохождения ограниченного по времени теста. Автоматика и Телемеханика, 1, 83-94. (In Russ.).
    Naumov A., Stepanov A., Ustinov A. (2024). On the Problem of Maximizing the Probability of Successful Passing of a Time-Limited Test. Automation and Remote Control.  1, pp.83-94 DOI: 10.31857/S0005231024010056
14. Хайрулин, Р.З. (2014) Моделирование развоза грузов по разветвленной сети автодорог. Вестник МГСУ,7, 184-191.
    Khairulin, R.Z. (2014).Modeling of cargo delivery along an extensive road network. Bulletin of MGSU (Russia),7, 184-191. (In Russ.).
15. Шестаков, А.В., Зуенко, А.А. (2022). Задачи логистики: классификация и методы решения. Труды Кольского научного центра РАН, Серия: Технические науки, 13 (2),  144–150. DOI: 10.37614/2949-1215.2022.13.2.014
    Shestakov, A.V., Zuenko, A.A. (2022). Logistics tasks: classification and solution methods. Proceedings of the Kola Scientific Center of the Russian Academy of Sciences, Series: Technical Sciences, 13 (2), 144-150. (In Russ.). DOI: 10.37614/2949-1215.2022.13.2.014
16. Kuravsky, L. S., Popkov, S. I. (2018). Forecasting macro parameters representing the behavior of an applied multi-agent system. International Journal of Modeling, Simulation, and Scientific Computing, 9(6), Art. 1850052. https://doi.org/10.1142/S1793962318500526
17. Naumov, A.V., Ustinov, A.E., Stepanov, A.E. (2024). On the problem of maximizing the probability of successful passing a time-limited test. Automation and Remote Control, 85(6), 60-67. https://doi.org/10.1134/S0005117924010053
18. Santoso, T., Ahmed, S., Goetschalckx, M., Shapiro, A. (2005). A stochastic programming approach for supply chain network design under uncertainty. European Journal of Operational Research, 167(1), 95–115. https://doi.org/10.18452/8297