Designing a Fuzzy Mathematical Model for a Two-Echelon Allocation-Routing Problem by Applying Route Conditions: A New Interactive Fuzzy Approach




Two-echelon allocation-routing model, Reliability, Multi-objective optimization, Interactive fuzzy approach


In vehicle routing problems (VRP), the optimal allocation of transportation by considering factors such as route hardness, driver experience and vehicle worn-out has a significant effect on costs reduction and approaching real-world conditions. In this paper, a novel fuzzy mixed integer non-linear mathematical model to address the two-echelon allocation-routing problem under uncertainty is proposed by applying route and fleet conditions. The cost of allocating drivers to diverse vehicles is computed at the first echelon of the problem, considering factors such as vehicle type, vehicle wear-out, and driver experience. Additionally, different routes are defused with varying levels of hardness. The goal of the second echelon of the model is to improve reliability by defining the reliability of routes within each section. To solve the model, the Torabi and Hessini (TH), the Selimi and Ozkarahan (SO) methods, and a newly proposed approach (PIA) were utilized to transform the multi-objective model into a single-objective one. Numerical tests and performance indicators were used to validate the effectiveness of both the multi-objective mathematical model and the proposed solution method. The validation computation results indicate that the proposed solution approach outperforms both the TH and SO approaches.


Download data is not yet available.


Farahbakhsh, A., & Kheirkhah, A. S. (2023). A new efficient genetic Algorithm-Taguchi-based approach for multi- period inventory routing problem. International journal of research in industrial engineering, 12(4), 397-413.

do C. Martins, L., Hirsch, P., & Juan, A. A. (2021). Agile optimization of a two‐echelon vehicle routing problem with pickup and delivery. International Transactions in Operational Research, 28(1), 201-221.

Movafaghpour, M. A. (2023). Developing an efficient algorithm for robust school bus routing with heterogeneous fleet. Journal of Decisions and Operations Research, 8(3), 566-577.

Yu, X., Zhou, Y., & Liu, X. F. (2020). The two-echelon multi-objective location routing problem inspired by realistic waste collection applications: The composable model and a metaheuristic algorithm. Applied Soft Computing, 94, 106477.

Cheng, C., Zhu, R., Costa, A. M., Thompson, R. G., & Huang, X. (2022). Multi-period two-echelon location routing problem for disaster waste clean-up. Transportmetrica A: Transport Science, 18(3), 1053-1083.

Chaube, S., Singh, S. B., Pant, S., & Kumar, A. (2018). Time-dependent conflicting bifuzzy set and its applications in reliability evaluation. Advanced Mathematical Techniques in Engineering Sciences, 4, 111-28.

Lagzaie, L., & Hamzehee, A. (2022). Providing a Multiproduct and Multiperiodic Model for Closed-Loop Green Supply Chain under Conditions of Uncertainty Based on a Fuzzy Approach for Solving Problem of Business Market. Complexity, 2022.

Wang, Y., Sun, Y., Guan, X., Fan, J., Xu, M., & Wang, H. (2021). Two-echelon multi-period location routing problem with shared transportation resource. Knowledge-Based Systems, 226, 107168.

Bahmani, V., Adibi, M. A., & Mehdizadeh, E. (2023). Integration of Two-Stage Assembly Flow Shop Scheduling and Vehicle Routing Using Improved Whale Optimization Algorithm. Journal of applied research on industrial engineering, 10(1).

Gandra, V. M. S., Çalık, H., Wauters, T., Toffolo, T. A., Carvalho, M. A. M., & Berghe, G. V. (2021). The impact of loading restrictions on the two-echelon location routing problem. Computers & Industrial Engineering, 160, 107609.

Fallahtafti, A., Ardjmand, E., Young Ii, W. A., & Weckman, G. R. (2021). A multi-objective two-echelon location-routing problem for cash logistics: A metaheuristic approach. Applied Soft Computing, 111, 107685.

Cao, J. X., Wang, X., & Gao, J. (2021). A two-echelon location-routing problem for biomass logistics systems. Biosystems Engineering, 202, 106-118.

Hajghani, M., Forghani, M. A., Heidari, A., Khalilzadeh, M., & Kebriyaii, O. (2023). A two-echelon location routing problem considering sustainability and hybrid open and closed routes under uncertainty. Heliyon, 9(3).

Khodashenas, M., Kazemipoor, H., Najafi, S. E., & Movahedi Sobhani, F. (2022). A two-stage uncertain model to arrange and locate vehicle routing with simultaneous pickup and delivery. International journal of research in industrial engineering, 11(3), 273-305.

Mohamed, I. B., Klibi, W., Sadykov, R., Şen, H., & Vanderbeck, F. (2023). The two-echelon stochastic multi-period capacitated location-routing problem. European Journal of Operational Research, 306(2), 645-667.

Xue, G., Wang, Y., Guan, X., & Wang, Z. (2022). A combined GA-TS algorithm for two-echelon dynamic vehicle routing with proactive satellite stations. Computers & Industrial Engineering, 164, 107899.

Du, J., Wang, X., Wu, X., Zhou, F., & Zhou, L. (2023). Multi-objective optimization for two-echelon joint delivery location routing problem considering carbon emission under online shopping. Transportation Letters, 15(8), 907-925.

Heidari, A., Imani, D. M., Khalilzadeh, M., & Sarbazvatan, M. (2023). Green two-echelon closed and open location-routing problem: application of NSGA-II and MOGWO metaheuristic approaches. Environment, Development and Sustainability, 25(9), 9163-9199.

Neira, D. A., Aguayo, M. M., De la Fuente, R., & Klapp, M. A. (2020). New compact integer programming formulations for the multi-trip vehicle routing problem with time windows. Computers & Industrial Engineering, 144, 106399.

Kumar, A., Vohra, M., Pant, S., & Singh, S. K. (2021). Optimization techniques for petroleum engineering: A brief review. International Journal of Modelling and Simulation, 41(5), 326-334.

Kumar, A., Pant, S., Ram, M., & Yadav, O. (Eds.). (2022). Meta-heuristic optimization techniques: applications in engineering (Vol. 10). Walter de Gruyter GmbH & Co KG.

Uniyal, N., Pant, S., Kumar, A., & Pant, P. (2022). Nature-inspired metaheuristic algorithms for optimization. Meta-heuristic Optimization Techniques, 1-10.

Kumar, A., Negi, G., Pant, S., Ram, M., & Dimri, S. C. (2021). Availability-cost optimization of butter oil processing system by using nature inspired optimization algorithms. Reliability: Theory & Applications, 16(SI 2 (64)), 188-200.

Kumar, A., Pant, S., Singh, M. K., Chaube, S., Ram, M., & Kumar, A. (2023). Modified Wild Horse Optimizer for Constrained System Reliability Optimization. Axioms, 12(7), 693.

Huang, N., Li, J., Zhu, W., & Qin, H. (2021). The multi-trip vehicle routing problem with time windows and unloading queue at depot. Transportation Research Part E: Logistics and Transportation Review, 152, 102370.

Rezaei Kallaj, M., Abolghasemian, M., Moradi Pirbalouti, S., Sabk Ara, M., & Pourghader Chobar, A. (2021). Vehicle routing problem in relief supply under a crisis condition considering blood types. Mathematical Problems in Engineering, 2021, 1-10.

Shiri, M., Ahmadizar, F., Thiruvady, D., & Farvaresh, H. (2023). A sustainable and efficient home health care network design model under uncertainty. Expert Systems with Applications, 211, 118185.

Wang, Y., Zhe, J., Wang, X., Sun, Y., & Wang, H. (2022). Collaborative multidepot vehicle routing problem with dynamic customer demands and time windows. Sustainability, 14(11), 6709.

Hasanpour Jesri, Z. S., Eshghi, K., Rafiee, M., & Van Woensel, T. (2022). The Multi-Depot Traveling Purchaser Problem with Shared Resources. Sustainability, 14(16), 10190.

Nozari, H., Tavakkoli-Moghaddam, R., & Gharemani-Nahr, J. (2022). A neutrosophic fuzzy programming method to solve a multi-depot vehicle routing model under uncertainty during the covid-19 pandemic. International Journal of Engineering, 35(2), 360-371.

Jiao, L., Peng, Z., Xi, L., Guo, M., Ding, S., & Wei, Y. (2023). A multi-stage heuristic algorithm based on task grouping for vehicle routing problem with energy constraint in disasters. Expert Systems with Applications, 212, 118740.

Pirabán-Ramírez, A., Guerrero-Rueda, W. J., & Labadie, N. (2022). The multi-trip vehicle routing problem with increasing profits for the blood transportation: An iterated local search metaheuristic. Computers & Industrial Engineering, 170, 108294.

Navazi, F., Tavakkoli-Moghaddam, R., Sazvar, Z., & Memari, P. (2019). Sustainable design for a bi-level transportation-location-vehicle routing scheduling problem in a perishable product supply chain. In Service Orientation in Holonic and Multi-Agent Manufacturing: Proceedings of SOHOMA 2018 (pp. 308-321). Springer International Publishing.

Asefi, A. H., Bozorgi-Amiri, A., & Ghezavati, V. (2020). Location-Routing Problem in Humanitarian Relief Chain Considering the Reliability of Road Network. Emergency Management, 9(1), 29-41.

Norouzi, N., Tavakkoli-Moghaddam, R., Ghazanfari, M., Alinaghian, M., & Salamatbakhsh, A. (2012). A new multi-objective competitive open vehicle routing problem solved by particle swarm optimization. Networks and Spatial Economics, 12, 609-633.

Al-Qudaimi, A., Kaur, K., & Bhat, S. (2021). Triangular fuzzy numbers multiplication: QKB method. Fuzzy Optimization and Modeling Journal, 2(2), 34-40. 10.30495/fomj.2021.1934118.1032

Liang, T. F. (2006). Distribution planning decisions using interactive fuzzy multi-objective linear programming. Fuzzy Sets and Systems, 157(10), 1303-1316.

Wang, R. C., & Liang, T. F. (2005). Applying possibilistic linear programming to aggregate production planning. International journal of production economics, 98(3), 328-341.

Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy sets and systems, 1(1), 45-55.

Lai, Y. J., & Hwang, C. L. (1992). A new approach to some possibilistic linear programming problems. Fuzzy sets and systems, 49(2), 121-133.

Selim, H., & Ozkarahan, I. (2008). A supply chain distribution network design model: an interactive fuzzy goal programming-based solution approach. The International Journal of Advanced Manufacturing Technology, 36, 401-418.

Torabi, S. A., & Hassini, E. (2008). An interactive possibilistic programming approach for multiple objective supply chain master planning. Fuzzy sets and systems, 159(2), 193-214.

Werners, B. M. (1988). Aggregation models in mathematical programming. In Mathematical models for decision support (pp. 295-305). Springer Berlin Heidelberg.

Diaz-Madronero, M., Peidro, D., & Mula, J. (2014). A fuzzy optimization approach for procurement transport operational planning in an automobile supply chain. Applied Mathematical Modelling, 38(23), 5705-5725.

Lai, Y. J., Hwang, C. L., Lai, Y. J., & Hwang, C. L. (1994). Fuzzy multiple objective decision making (pp. 139-262). Springer Berlin Heidelberg.



How to Cite

Yadegari, Z., Hadji Molana, S. M., Husseinzadeh Kashan, A., & Najafi, S. E. (2024). Designing a Fuzzy Mathematical Model for a Two-Echelon Allocation-Routing Problem by Applying Route Conditions: A New Interactive Fuzzy Approach. Decision Making: Applications in Management and Engineering, 7(2), 172–196.