Modeling the impact of uncertain demand on intracity express delivery network design by adjustable data-driven robust approach

Shijun Wang; Yanjiao Wang; Naiqi Liu; Weida Zhang; Shijun Wang; Yanjiao Wang; Naiqi Liu; Weida Zhang

doi:10.3934/jimo.2026055

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 3: 1491-1518. doi: 10.3934/jimo.2026055

Previous Article Next Article

Research article

Modeling the impact of uncertain demand on intracity express delivery network design by adjustable data-driven robust approach

1.
School of Law and Public Administration, Hebei Normal University, Shijiazhuang 050024, Hebei, China
2.
Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics & Information Science, Hebei University, Baoding 071002, Hebei, China
3.
College of Management and Economics, Tianjin University, 300072, Tianjin, China

Received: 05 January 2026 Revised: 29 January 2026 Accepted: 05 February 2026 Published: 27 February 2026
90B06, 90C17

With the rapid development of e-commerce, intracity express delivery services have become an integral component of modern logistics systems while facing unprecedented challenges. This paper addresses the service network design problem for capacity-constrained intracity express delivery systems. We propose a two-stage decision framework wherein the first stage determines vehicle routing plans and calculates transportation costs, and the second stage arranges parcel transportation based on actual demand and evaluates the associated penalty costs. Recognizing the uncertainty in demand inherent to real-world operations, this paper specifically examines how demand fluctuations affect system performance. To address this challenge, we construct an uncertainty set using support vector clustering (SVC) based on available historical data, which serves as the foundation for developing a data-driven adjustable robust optimization (ARO) model. Through linear decision rules and conic optimization theory, we transform the robust model into an equivalent mixed-integer linear programming formulation and develop a customized Benders dual decomposition algorithm for its solution. The proposed methodology is then applied to a case study concerning an autonomous delivery vehicle service network design in Yuhang, China. Computational results validate the effectiveness of our optimization approach.
Citation: Shijun Wang, Yanjiao Wang, Naiqi Liu, Weida Zhang. Modeling the impact of uncertain demand on intracity express delivery network design by adjustable data-driven robust approach[J]. Journal of Industrial and Management Optimization, 2026, 22(3): 1491-1518. doi: 10.3934/jimo.2026055

Related Papers:

Abstract

With the rapid development of e-commerce, intracity express delivery services have become an integral component of modern logistics systems while facing unprecedented challenges. This paper addresses the service network design problem for capacity-constrained intracity express delivery systems. We propose a two-stage decision framework wherein the first stage determines vehicle routing plans and calculates transportation costs, and the second stage arranges parcel transportation based on actual demand and evaluates the associated penalty costs. Recognizing the uncertainty in demand inherent to real-world operations, this paper specifically examines how demand fluctuations affect system performance. To address this challenge, we construct an uncertainty set using support vector clustering (SVC) based on available historical data, which serves as the foundation for developing a data-driven adjustable robust optimization (ARO) model. Through linear decision rules and conic optimization theory, we transform the robust model into an equivalent mixed-integer linear programming formulation and develop a customized Benders dual decomposition algorithm for its solution. The proposed methodology is then applied to a case study concerning an autonomous delivery vehicle service network design in Yuhang, China. Computational results validate the effectiveness of our optimization approach.

References

[1]	I. Dayarian, A. Rocco, A. Erera, M. Savelsbergh, Operations design for high-velocity intra-city package service, Transp. Res. Part B Methodol., 161 (2022), 150–168. https://doi.org/10.1016/j.trb.2022.05.002 doi: 10.1016/j.trb.2022.05.002
[2]	Y. H. Wu, H. Fang, A. G. Qureshi, T. Yamada, Capacitated hub location routing problem with time windows and stochastic demands for the design of intra-city express systems, Eur. J. Oper. Res., 326 (2025), 255–269. https://doi.org/10.1016/j.ejor.2025.05.006 doi: 10.1016/j.ejor.2025.05.006
[3]	X. P. Wang, J. L. Zhao, Distributionally robust optimization of the vehicle routing problem with uncertain customers, J. Ind. Manag. Optim., 21 (2025), 1983–2006. https://doi.org/10.3934/jimo.2024159 doi: 10.3934/jimo.2024159
[4]	State post bureau of the People's Republic of China, 2024 Postal industry development statistical bulletin, 2025. Available from: https://www.spb.gov.cn/gjyzj/c100276/202501/460d02f2e54c4d0ebbba3a4f431d0042.shtml.
[5]	C. Liu, J. J. Zhou, J. W. Gan, Y. X. Wu, Y. L. Huang, J. H. Shao, et al., Optimizing the ground intra-city express delivery network: An integrated multiple centrality assessment, multi-criteria decision-making, and multi-objective integer programming model, J. Intell. Transp. Syst., 28 (2024), 525–543. https://doi.org/10.1080/15472450.2022.2157211 doi: 10.1080/15472450.2022.2157211
[6]	X. Chu, S. W. Chen, K. Wang, L. X. Wu, G. Y. Xu, A cost-efficiency analysis of drones in revolutionizing intra-city express services, Adv. Eng. Inform., 65 (2025), 103324. https://doi.org/10.1016/j.aei.2025.103324 doi: 10.1016/j.aei.2025.103324
[7]	M. Seifbarghy, F. Yargholi, M. Hamidi, Modeling a stochastic multi-objective, multi-site, multi-mode, and multi-item supplier selection and order allocation problem, J. Ind. Manag. Optim., 20 (2024), 2980–3003. https://doi.org/10.3934/jimo.2024037 doi: 10.3934/jimo.2024037
[8]	R. Diaz, C. Phan, D. Golenbock, B. Sanford, A prescriptive framework to support express delivery supply chain expansions in highly urbanized environments, Ind. Manag. Data Syst., 122 (2022), 1707–1737. https://doi.org/10.1108/IMDS-02-2022-0076 doi: 10.1108/IMDS-02-2022-0076
[9]	L. B. Guo, X. T. Guo, Y. J. Wang, Decision-making for demand signal in crowd sourcing platform, J. Ind. Manag. Optim., 20 (2024), 664–678. https://doi.org/10.3934/jimo.2023096 doi: 10.3934/jimo.2023096
[10]	Y. J. Wang, A. X. Chen, N. Q. Liu, Constructing resilient supply chain for risk-averse buyers by data-driven robust optimization approach, Int. J. Prod. Econ., (2025), 109734. https://doi.org/10.1016/j.ijpe.2025.109734 doi: 10.1016/j.ijpe.2025.109734
[11]	H. Mollashahi, Optimization of closed-loop supply chains with a waste management and recycling approach: examining the effects of capacity, facility location, and uncertainty management in responding to variable demand, J. Ind. Manag. Optim., 21 (2025), 5435–5453. https://doi.org/10.3934/jimo.2025099 doi: 10.3934/jimo.2025099
[12]	R. P. Huang, S. J. Qu, Z. M. Liu, Two-stage distributionally robust optimization model for warehousing-transportation problem under uncertain environment, J. Ind. Manag. Optim., 19 (2023), 6344–6363. https://doi.org/10.3934/jimo.2022218 doi: 10.3934/jimo.2022218
[13]	Q. Meng, X. L. Hei, S. A. Wang, H. J. Mao, Carrying capacity procurement of rail and shipping services for automobile delivery with uncertain demand, Transp. Res. Part E Logist. Transp. Rev., 82 (2015), 38–54. https://doi.org/10.1016/j.tre.2015.07.005 doi: 10.1016/j.tre.2015.07.005
[14]	S. Khalifehzadeh, M. B. Fakhrzad, Y. Z. Mehrjerdi, H. Hosseini_Nasab, Two effective metaheuristic algorithms for solving a stochastic optimization model of a multi-echelon supply chain, Appl. Soft Comput., 76 (2019), 545–563. https://doi.org/10.1016/j.asoc.2018.12.018 doi: 10.1016/j.asoc.2018.12.018
[15]	M. Q. Yin, M. Huang, X. W. Wang, L. H. Lee, Fourth-party logistics network design under uncertainty environment, Comput. Ind. Eng., 167 (2022), 108002. https://doi.org/10.1016/j.cie.2022.108002 doi: 10.1016/j.cie.2022.108002
[16]	D. Petrovic, M. Kalata, J. Luo, A fuzzy scenario-based optimisation of supply network cost, robustness and shortages, Comput. Ind. Eng., 160 (2021), 107555. https://doi.org/10.1016/j.cie.2021.107555 doi: 10.1016/j.cie.2021.107555
[17]	Y. Sun, Fuzzy approaches and simulation-based reliability modeling to solve a road–rail intermodal routing problem with soft delivery time windows when demand and capacity are uncertain, Int. J. Fuzzy Syst., 22 (2020), 2119–2148. https://doi.org/10.1007/s40815-020-00905-x doi: 10.1007/s40815-020-00905-x
[18]	J. Q. Guo, L. Chen, Configuration and optimisation of a green closed-loop supply chain with delivery time and green investment considering government subsidy under meta-heuristics algorithms, Int. J. Syst. Sci. Oper. Logist., 11 (2024), 2401033. https://doi.org/10.1080/23302674.2024.2401033 doi: 10.1080/23302674.2024.2401033
[19]	X. Xiang, T. Fang, C. C. Liu, Z. Pei, Robust service network design problem under uncertain demand, Comput. Ind. Eng., 172 (2022), 108615. https://doi.org/10.1016/j.cie.2022.108615 doi: 10.1016/j.cie.2022.108615
[20]	E. Taniguchi, M. Noritake, T. Yamada, T. Izumitani, Optimal size and location planning of public logistics terminals, Transp. Res. Part E Logist. Transp. Rev., 35 (1999), 207–222. https://doi.org/10.1016/S1366-5545(99)00009-5 doi: 10.1016/S1366-5545(99)00009-5
[21]	J. Nicholson, F. Gzara, A. Alnaggar, Unmanned aerial vehicle traffic network design with risk mitigation, Transp. Res. Part E Logist. Transp. Rev., 204 (2025), 104380. https://doi.org/10.1016/j.tre.2025.104380 doi: 10.1016/j.tre.2025.104380
[22]	O. Satici, I. Dayarian, Tactical and operational planning of express intra-city package services, Omega, 122 (2024), 102940. https://doi.org/10.1016/j.omega.2023.102940 doi: 10.1016/j.omega.2023.102940
[23]	X. Wang, L. Wang, C. X. Dong, H. Ren, K. Xing, An online deep reinforcement learning-based order recommendation framework for rider-centered food delivery system, IEEE Trans. Intell. Transp. Syst., 24 (2023), 5640–5654. https://doi.org/10.1109/TITS.2023.3237580 doi: 10.1109/TITS.2023.3237580
[24]	A. Behrendt, M. Savelsbergh, H. Wang, A prescriptive machine learning method for courier scheduling on crowdsourced delivery platforms, Transp. Sci., 57 (2023), 889–907. https://doi.org/10.1287/trsc.2022.1152 doi: 10.1287/trsc.2022.1152
[25]	A. Ben-Hur, D. Horn, H. T. Siegelmann, V. Vapnik, Support vector clustering, J. Mach. Learn. Res., 2 (2001), 125–137.
[26]	C. Shang, X. L. Huang, F. Q. You, Data-driven robust optimization based on kernel learning, Comput. Chem. Eng., 106 (2017), 464–479. https://doi.org/10.1016/j.compchemeng.2017.07.004 doi: 10.1016/j.compchemeng.2017.07.004
[27]	A. Ben-Tal, L. El Ghaoui, A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009.
[28]	M. Bodur, J. R. Luedtke, Two-stage linear decision rules for multi-stage stochastic programming, Math. Program., 191 (2022), 347–380. https://doi.org/10.1007/s10107-018-1339-4 doi: 10.1007/s10107-018-1339-4
[29]	S. J. Garstka, R. J. B. Wets, On decision rules in stochastic programming, Math. Program., 7 (1974), 117–143. https://doi.org/10.1007/BF01585511 doi: 10.1007/BF01585511
[30]	R. Rahmaniani, S. Ahmed, T. G. Crainic, M. Gendreau, W. Rei, The Benders dual decomposition method, Oper. Res., 68 (2020), 878–895. https://doi.org/10.1287/opre.2019.1892 doi: 10.1287/opre.2019.1892
[31]	S. J. Wang, Y. Lu, F. Chu, J. B. Yu, Scheduling with divisible jobs and subcontracting option, Comput. Oper. Res., 145 (2022), 105850. https://doi.org/10.1016/j.cor.2022.105850 doi: 10.1016/j.cor.2022.105850
[32]	A. G. Lium, T. G. Crainic, S. W. Wallace, A study of demand stochasticity in service network design, Transp. Sci., 43 (2009), 144–157. https://doi.org/10.1287/trsc.1090.0265 doi: 10.1287/trsc.1090.0265

jimo-22-03-055-s001.pdf

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)