MODEL AND METHOD FOR SOLVING THE TRANSPORT PROBLEM WITH A THRESHOLD CHANGE IN TRANSPORTATION TARIFFS
https://doi.org/10.33815/2313-4763.2026.1.32.164-173
Abstract
The article presents a transport problem with a threshold change in transportation tariffs, in which the cost of transportation depends on the volume of cargo between specific suppliers and consumers. The relevance of the study is due to the spread of flexible tariff mechanisms in modern transport and logistics systems, under which reduced or wholesale tariffs are applied when a certain volume of transportation is exceeded. Similar approaches are used in cargo consolidation systems, long-term contracts, multimodal transportation and in organizing regular deliveries of large batches of products. A mathematical model of the transport problem is proposed, in which for each “supplier-consumer” pair basic and reduced transportation tariffs are set, as well as threshold values of cargo volumes, after which the alternative delivery cost is applied. The objective function of the problem is constructed taking into account the dynamic change in transportation costs depending on the volume of transport flow and the system of balance constraints of the transport problem. A modified method for constructing a reference plan based on the classical least cost method has been developed, which takes into account the possibility of switching to reduced tariffs when reaching transportation threshold values. A modified potential method has also been proposed for finding the optimal plan, which provides for repeated updating of the effective cost matrix and recalculation of potentials after each cargo redistribution. Unlike the classical transport problem, in the proposed model the structure of transport costs can change directly in the process of forming a transportation plan. The practical significance of the study lies in the possibility of using the developed model to improve the efficiency of freight transportation management in the conditions of wholesale tariffs and dynamic changes in the cost of delivery.
References
2. Slavych, V. P., Biloushchenko, D. Yu. (2025). Model of the transport problem in the case of cargo delivery by two different types of vehicles. Naukovyi visnyk KhDMA. No 31. P. 178–186. http://doi.org/10.33815/2313-4763.2025.2.31.178-186.
3. Bera, R. K., Mondal, S. K. (2022). A multi-objective transportation problem with cost dependent credit period policy under Gaussian fuzzy environment. Operational Research. 22(4): 3147–3182. http://doi.org/10.1007/s12351-022-00691-4.
4. Bera, R. K., Mondal, S. K. (2023). A multi-objective transportation problem under quantity dependent credit period and cost structure policies in triangular intuitionistic fuzzy environment. Engineering Applications of Artificial Intelligence. 123: 106396. http://doi.org/10.1016/j.engappai.2023.106396.
5. Gütmen, S., Roy, S. K., Weber, G. W. (2024). An overview of weighted goal programming: a multi-objective transportation problem with some fresh viewpoints. Central European Journal of Operations Research. 32(3): 557–568. http://doi.org/10.1007/s10100-023-00861-5.
6. Luo, R., Ji, S., Ji, T. (2021). An effective chaos-driven differential evolution for multi-objective unbalanced transportation problem considering fuel consumption. Applied Soft Computing. 101: 107058. http://doi.org/10.1016/j.asoc.2020.107058.
7. Slavych, V. P. et al. (2025). Mathematical and information model of a special type of transport task as a tool for increasing economic efficiency of freight transportation. Artificial intelligence as a tool to protect the economy from disinformation: innovative solutions and international practices. Riga, Latvia: Baltija Publishing. 258 p. http://doi.org/10.30525/978-9934-26-586-0.
8. Mondal, M., Mandal, G., Shaikh, A. A., & Bhunia, A. K. (2025). Investigation of a multi-objective fixed charge transportation problem with quantity dependent transportation cost and discount policy via metaheuristics. INFOR: Information Systems and Operational Research, 63(3), 562–591. https://doi.org/10.1080/03155986.2025.2492739.
9. Ghosh, S., Küfer, K. H., Roy, S. K., Weber, G. W. (2023). Type-2 zigzag uncertain multi-objective fixed-charge solid transportation problem: time window vs. preservation technology. Central European Journal of Operations Research. 31(1): 337–362. http://doi.org/10.1007/s10100-022-00811-7.
10. Ghosh, S., Roy, S. K. (2021). Fuzzy-rough multi-objective product blending fixed-charge transportation problem with truck load constraints through transfer station. RAIRO-Oper Res. 55: S2923–S2952. http://doi.org/10.1051/ro/2020129.
11. Ghosh, S., Roy, S. K., Verdegay, J. L. (2022). Fixed-charge solid transportation problem with budget constraints based on carbon emission in neutrosophic environment. Soft Computing. 26(21): 11611–11625. http://doi.org/10.1007/s00500-022-07442-9.
12. Haque, S., Bhurjee, A. K., Kumar, P. (2022). Multi-objective non-linear solid transportation problem with fixed charge, budget constraints under uncertain environments. Systems Science & Control Engineering. 10(1): 899–909. http://doi.org/10.1080/21642583.2022.2137707.
13. Keshavarz E., Mahmoodirad A., Niroomand S. (2023). A transportation problem considering fixed charge and fuzzy shipping costs. Decision Making Advances . 1(1): 115–122. http://doi.org/10.31181/dma11202313.
14. Mahmoodirad, A., Niroomand, S. (2023). A heuristic approach for fuzzy fixed charge transportation problem. Journal of Decision Analytics and Intelligent Computing. 3(1): 139–147. http://doi.org/10.31181/jdaic10005092023m.
15. Mondal, A., Roy, S. K., Midya, S. (2023). Intuitionistic fuzzy sustainable multi-objective multi-item multi-choice step fixed-charge solid transportation problem. Journal of Ambient Intelligence and Humanized Computing. 14(6): 6975–6999. http://doi.org/10.1007/s12652-021-03554-6.
