基于分布鲁棒优化的无人机配送设施选址分配问题

doi:10.13229/j.cnki.jdxbgxb.20240868

Abstract

Abstract:

This paper focuses on the urban last-mile delivery process under the drone service model， optimizing drone delivery facility location and service allocation strategies through a two-stage mathematical programming model. The impact of demand uncertainty on location-allocation decisions is characterized using a distributionally robust optimization method. The original model is equivalently transformed into a mixed-integer second-order cone programming problem， and an outer approximation algorithm is proposed to improve solving efficiency. Simulation verification is conducted based on the pharmaceutical delivery data in Songjiang District， Shanghai. The results indicate that the proposed outer approximation algorithm can reduce the computational time of commercial solvers by 32.07%； compared with the deterministic model， the proposed distributionally robust optimization model can increase the system's total profit by 4.21 times when demand is highly volatile.

Key words: transportation planning and management, drone delivery, facility location-allocation, demand uncertainty, distributionally robust optimization, outer approximation, low-altitude economy

CLC Number:

F252.3

Kang-lin LIU,Ze-yu ZHANG,Jing-wen JIANG,Xun GONG,Yao CHEN. Distributionally robust optimization for drone delivery facility location and allocation problem[J].Journal of Jilin University(Engineering and Technology Edition), 2026, 56(2): 464-472.

Figures/Tables 10

Fig.1

Table 1

Explanation of symbols"

类型	符号	含义
参数	$m$	备选设施总数
	$n$	需求点总数
	$r$	无人机总数
	$I$	所有备选配送设施位置的集合， $i ∈ I, i = 1,2, ?, m$
	$J$	所有需求点的集合， $j ∈ J, j = 1,2, ?, n$
	$K$	所有可用无人机的集合， $k ∈ K, k = 1,2, ?, r$
	$L$	每架无人机的最大飞行里程，单位为km
	$R$	每套配送设施允许配备无人机的最大数量
	$P$	允许选定配送设施的最大数量
	q_j	需求点 $j$ 的需求量，单位为kg/d
	$l i j$	往返配送设施 $i$ 和需求点 $j$ 之间的距离，单位为km
	$c i$	配送设施 $i$ 的运营成本，单位为元/d
	$c 0$	每架无人机的使用成本，单位为元/d
	$θ$	满足单位需求所产生的收益，单位为元/kg
	$η$	配送服务覆盖率
决策变量	$x i$	若选定配送设施 $i$ ，则取1
	y_ij	若需求点 $j$ 分配给配送设施 $i$ 服务，则取1
	$z i k$	若无人机 $k$ 分配给配送设施 $i$ ，则取1
	$w i j k$	若配送设施 $i$ 中的无人机 $k$ 对需求点 $j$ 进行配送，则取1

Table 1

Fig.2

Table 2

Table 3

Fig.3

Fig.4

Table 4

Table 5

Sensitivity analysis of the safety factor changes"

$λ$	有效最大飞行里程/km	需求覆盖率 $η$ /%	总利润/元
1.00	12.500	93.94	257.0
1.05	11.875	90.91	228.0
1.11	11.250	87.88	202.0
1.18	10.625	84.85	191.0
1.25	10.000	81.82	167.0

Table 5

Fig.5

References 16

[1]	张世昌, 许芳菲. 低空经济生成与发展的伦理考量——基于马克思空间思想的视角[J]. 北京航空航天大学学报:社会科学版, 2024, 37(6): 1-9.
	Zhang Shi-chang, Xu Fang-fei. Ethical considerations on the generation and development of low-altitude economy: a perspective based on marx's spatial[J]. Journal of Beihang University (Social Sciences Edition), 2024, 37(6): 1-9.
[2]	何颖, 樊俊松, 王巍, 等. 无人机空地安全通信与航迹规划的多目标联合优化方法[J]. 吉林大学学报:工学版, 2023, 53(3): 913-922.
	He Ying, Fan Jun-song, Wang Wei, et al. Joint optimization of secure communication and trajectory planning in unmanned aerial vehicle air⁃to⁃ground[J]. Journal of Jilin University (Engineering and Technology Edition), 2023, 53(3): 913-922.
[3]	陈明舟. 无人机物流配送服务动向[J]. 物流工程与管理, 2020, 42(10): 90-92.
	Chen Ming-zhou. Trends in logistics delivery services using UAV[J]. Logistics Engineering and Management, 2020, 42(10): 90-92.
[4]	陈刚, 付江月. 军民融合背景下无人机配送中心选址问题研究[J]. 计算机工程与应用, 2019, 55(8): 226-231.
	Chen Gang, Fu Jiang-yue. Research on the location problem of drone delivery centers under the background of military-civilian integration[J]. Computer Engineering and Applications, 2019, 55(8): 226-231.
[5]	Golabi M, Shavarani S M, Izbirak G. An edge-based stochastic facility location problem in UAV supported humanitarian relief logistics: a case study of Tehran earthquake[J]. Natural Hazards, 2017, 87(3):1545-1565.
[6]	Hong I, Kuby M, Murray A T. A range-restricted recharging station coverage model for drone delivery service planning[J]. Transportation Research Part C: Emerging Technologies, 2018, 90:198-212.
[7]	钱欣悦, 张洪海, 张芳,等. 末端配送物流无人机起降点选址分配问题研究[J].武汉理工大学学报:交通科学与工程版, 2021, 45(4): 682-687.
	Qian Xin-yue, Zhang Hong-hai, Zhang Fang, et al. Research on location allocation of UAV landing points for terminal distribution logistics[J]. Journal of Wuhan University of Technology (Transportation Science & Engineering), 2021, 45(4): 682-687.
[8]	Chauhan D, Unnikrishnan A, Figliozzi M. Maximum coverage capacitated facility location problem with range constrained drones[J]. Transportation Research Part C: Emerging Technologies, 2019, 99:1-18.
[9]	李艳波, 柳柏松, 姚博彬, 等. 考虑路网随机特性的高速公路换电站选址[J]. 吉林大学学报:工学版, 2023, 53(5): 1364-1371.
	Li Yan-bo, Liu Bai-song, Yao Bo-bin, et al. Location of electrical changing station of expressway considering stochastic characteristics of road network[J]. Journal of Jilin University (Engineering and Technology Edition), 2023, 53(5): 1364-1371.
[10]	赖志柱, 王铮, 戈冬梅, 等. 多目标应急物流中心选址的鲁棒优化模型[J]. 运筹与管理, 2020, 29(5): 74-83.
	Lai Zhi-zhu, Wang Zheng, Ge Dong-mei, et al. A multi-objective robust optimization model for emergency logistics center location[J]. Operations Research and Management Science, 2020, 29(5): 74-83.
[11]	鲜斌, 王光怡, 蔡佳明. 多无人机吊挂负载运输系统的非线性鲁棒控制设计[J]. 吉林大学学报: 工学版, 2024, 54(6): 1788-1795.
	Xian Bin, Wang Guang-yi, Cai Jia-ming. Nonlinear robust control design for multi unmanned aerial vehicles suspended payload transportation system[J]. Journal of Jilin University (Engineering and Technology Edition), 2024, 54(6): 1788-1795.
[12]	何勇, 张成义, 李姗姗. 基于两阶段鲁棒优化的无人机载机平台调度问题[J].系统工程学报, 2020, 35(6): 838-848.
	He Yong, Zhang Cheng-yi, Li Shan-shan. Unmanned aerial vehicle carriers scheduling problem based on two-stage robust optimization[J]. Journal of Systems Engineering, 2020, 35(6): 838-848.
[13]	Zhu T K, Boyles S D, Unnikrisshnan A. Two-stage robust facility location problem with drones[J]. Transportation Research Part C: Emerging Technologies, 2022, 137: 103563.
[14]	Zhang Y L, Shen Z J M, Song S J. Distributionally robust optimization of two-Stage lot-Sizing problems[J]. Production and Operations Management, 2016, 25(12): 2116-2131.
[15]	Delage E, Ye Y Y. Distributionally robust optimization under moment uncertainty with application to data-driven problems[J]. Operations Research, 2010, 58 (3):595-612.
[16]	郑雨欣. 城市物流无人机运行中若干关键问题研究[D]. 天津: 中国民航大学空中交通管理学院, 2022.
	Zheng Yu-xin. Research on several key issues in urban logistics drone operations[D]. Tianjin: College of Air Traffic Management, Civil Aviation University of China, 2022.

Related Articles 15

[1]	Yi ZHANG,Sha-wen CHEN,Yan CHEN,Xiang-yu FAN,Si-qi WANG,Huan WU,Peng-fei JIAO. Dynamic operation and maintenance evaluation and predictive maintenance of mechanical and electrical equipment on highways [J]. Journal of Jilin University(Engineering and Technology Edition), 2026, 56(1): 170-182.
[2]	Yao SUN,Dong-xuan BAI,Bao-zhen YAO,Zi-jian BAI. Characteristics analysis of port and city transportation network based on percolation theory [J]. Journal of Jilin University(Engineering and Technology Edition), 2026, 56(1): 199-208.
[3]	Hong-fei JIA,Bo ZHUANG,Qing-yu LUO,Ling LIU,Qiu-yang HUANG. Decision optimization of urban road network performance restoration under waterlogged conditions [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(9): 2969-2977.
[4]	Dong WANG,Yu-xuan LI,Huan WU,Fang ZONG. Rating algorithm for open test road of intelligent connected vehicle based on random forest [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(9): 2998-3006.
[5]	Qing-ling HE,Yu-long PEI,Lin HOU,Jing LIU,Sheng PAN. Hybrid strategy improves WOA⁃BiLSTM speed prediction of expressway exit ramp [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(6): 2038-2049.
[6]	Hui-zhi XU,Dong-sheng HAO,Xiao-ting XU,Shi-sen JIANG. Expressway small object detection algorithm based on deep learning [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(6): 2003-2014.
[7]	Hao YUE,Xiao CHANG,Jian-ye LIU,Qiu-shi QU. Customized bus route optimization with vehicle window [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(4): 1266-1274.
[8]	Xiang-hai MENG,Guo-rui WANG,Ming-yang ZHANG,Bi-jiang TIAN. Traffic accident prediction model of mountain highways based on selection integration [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(4): 1298-1306.
[9]	Wen-jing WU,Chun-chun DENG,Hong-fei JIA,Shu-hang SUN. Evaluation of road network unblocked reliability and identification of critical sections under influence of flooding [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(4): 1250-1257.
[10]	Yu-ran LI,Fei WANG,Cai-hua ZHU,Fei HAN,Yan LI. Chain-effect utility of factors influencing residents' commuting mode choice in polluted weather [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(2): 577-590.
[11]	Tian-yang GAO,Da-wei HU,Rui-sen JIANG,Xue WU,Hui-tian LIU. Optimization study of zonal-based flexible feeder bus routes based on modular vehicle system [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(2): 537-545.
[12]	Shu-hong MA,Jun-jie ZHANG,Xi-fang CHEN,Guo-mei LIAO. Identifying urban functional structures using time-series taxi data [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(2): 603-613.
[13]	Da-yi QU,Shou-chen DAI,Yi-cheng CHEN,Shan-ning CUI,Yu-xiang YANG. Modeling of vehicle game cut-out and merging behavior based on trajectory data [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(10): 3208-3220.
[14]	Hui-zhi XU,Shi-sen JIANG,Xiu-qing WANG,Shuang CHEN. Vehicle target detection and ranging in vehicle image based on deep learning [J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(1): 185-197.
[15]	Chang-jiang ZHENG,Tong-tong TAO,Zhi-chao CHEN. Cascading failure model based on adjustable redistribution of traffic flow [J]. Journal of Jilin University(Engineering and Technology Edition), 2024, 54(9): 2441-2450.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed

Comments

Recommended 0

No Suggested Reading articles found!

规模参数			求解器		遗传算法		OAM
m	n	r	求解时间均值	求解时间准差	求解时间均值	求解结果/元	求解时间均值	求解时间标准差	求解结果/元
20	10	5	1.32	0.12	0.31	45.24	0.67	0.05	96.23
30	10	5	1.61	0.06	0.78	56.21	1.28	0.06	69.32
30	20	5	34.12	2.25	1.84	121.57	28.71	1.22	143.85
40	20	5	173.27	1.07	39.95	6.52	103.49	1.44	8.83
40	20	10	777.07	8.22	585.6	-9.14	598.65	5.36	33.89
40	30	10	1 135.53	12.65	114.77	6.68	945.67	8.31	8.91
50	20	10	2 318.81	31.46	605.65	-52.04	1 079.11	11.25	26.68
60	30	10	1 910.47	27.65	603.21	-3.02	1 473.86	16.78	81.43

分布鲁棒模型			确定性模型
选定设施序号	配备无人机的数目	无人机所服务的需求点	选定设施序号	配备无人机的数目	无人机所服务的需求点
2	3	无人机11：1	2	3	无人机1：3
		无人机2：4、14			无人机2：4、14
		无人机3：6、8			无人机3：6、8
10	3	无人机4：2、10	3	3	无人机4：1、7
		无人机5：12			无人机5：16
		无人机6：7、16			无人机6：2、10
20	3	无人机7：7	20	3	无人机7：5
		无人机8：11、17			无人机8：11、17
		无人机9：18、20			无人机9：18、20

模型类型	配送利润/元	需求覆盖率/%	总利润/元
确定性模型	423.54	75	6.54
分布鲁棒模型	444.55	75	27.55

Distributionally robust optimization for drone delivery facility location and allocation problem

RICH HTML

PDF (PC)