基于硬时间窗的空中加油任务分配优化方法

doi:10.12305/j.issn.1001-506X.2026.07.20

摘要/Abstract

摘要：

针对防御性制空作战中多个巡逻阵位之间的多加油机协同保障任务规划问题，提出一种基于硬时间窗约束的空中加油任务分配优化方法。首先，通过分析战斗空中巡逻任务的特性，构建以最小化加油机出动架次为核心并兼顾等待成本与燃油利用效率的空中加油任务分配优化模型。其次，针对该问题的特点设计改进的自适应多阶段蚁群优化算法，通过动态调整启发式信息因子与信息素更新机制，增强算法对硬时间窗约束的适应性与全局搜索能力。最后，将计算得到的优化方案结果在成本优化上与原始规划方案进行比较，并结合大规模算例对该算法的时间复杂度、空间复杂度进行了讨论，明确了该算法在处理大规模空中加油任务分配时的优势与局限。

关键词: 空中加油, 硬时间窗, 任务分配, 蚁群优化算法

Abstract:

For the mission planning problem of collaborative multi-tanker support to multiple patrol formations in defensive air superiority operations, an air refueling mission assignment optimization method based on hard time window constraints is proposed. Firstly, by analyzing the characteristics of combat air patrol missions, an air refueling mission assignment optimization model is constructed with the core objective of minimizing tanker sortie while considering waiting cost and fuel utilization efficiency. Secondly, an improved adaptive multi-stage ant colony optimization algorithm is designed. By dynamically adjusting heuristic information factors and pheromone update mechanisms, the algorithm enhances adaptability to hard time window constraints and global search capability. Finally, the optimized plan result is compared with the original assignment scheme in terms of cost reduction. Through large-scale numerical experiments, the algorithm’s time and space complexity are analyzed, elucidating its advantages and limitations for large-scale air refueling mission assignment.

Key words: air refueling, hard time window, task assignment, ant colony optimization (ACO) algorithm

中图分类号:

V 271.4+94

马旭, 余付平, 沈堤. 基于硬时间窗的空中加油任务分配优化方法[J]. 系统工程与电子技术, 2026, 48(7): 2352-2366.

Xu MA, Fuping YU, Di SHEN. Optimization method of air refueling mission assignment based on hard time window[J]. Systems Engineering and Electronics, 2026, 48(7): 2352-2366.

图/表 22

图1

图2

图3

图4

图5

表1

表2

表3

图6

图7

图8

图9

图10

图11

图12

图13

表4

图14

表5

图15

图16

图17

参考文献 35

1	徐鑫宇, 万路军, 陈平, 等. 防御性制空截击中警戒巡逻空域规划建模[J]. 系统工程与电子技术, 2022, 44 (5): 1589- 1599. doi: 10.12305/j.issn.1001-506X.2022.05.20
	XU X Y, WAN L J, CHEN P, et al. Modeling of alert patrol airspace planning in defensive counter air interception operation[J]. Systems Engineering and Electronics, 2022, 44 (5): 1589- 1599. doi: 10.12305/j.issn.1001-506X.2022.05.20
2	吴傲, 李寰宇, 高杨军, 等. 面向巡逻阵位保持的空中加油管理时空协同方法[J]. 航空学报, 2023, 44 (20): 628438. doi: 10.7527/S1000-6893.2023.28438
	WU A, LI H Y, GAO Y J, et al. Time space coordination method for air refueling management for combat air patrol position maintenance[J]. Acta Aeronautica et Astronautica Sinica, 2023, 44 (20): 628438. doi: 10.7527/S1000-6893.2023.28438
3	陈世康, 王智民, 梁标, 等. 面向体系仿真的蓝军舰艇编队防空作战指控建模方法[J]. 系统仿真学报, 2024, 36 (10): 2246- 2256. doi: 10.16182/j.issn1004731x.joss.24-0865
	CHEN S K, WANG Z M, LIANG B, et al. Modeling method of blue army warship formation air defense command and control for system-of-systems simulation[J]. Journal of System Simulation, 2024, 36 (10): 2246- 2256. doi: 10.16182/j.issn1004731x.joss.24-0865
4	刘佳铭, 李熠, 李海旭. 美军舰载航空作战任务与出动模式分析[J]. 船舶工程, 2021, 43 (S2): 33- 37，52.
	LIU J M, LI Y, LI H X. Analysis of operational mission and dispatch mode of U. S. ship-borne aviation[J]. Ship Engineering, 2021, 43 (S2): 33- 37，52.
5	JORDAO E B. An investigation of the combat air patrol stationing in an integrated air defense scenario[D]. Monterey: Naval Postgraduate School, 1990.
6	李前坤, 李飞, 郭振. 基于人工智能的空战复杂性研究[J]. 军事文摘, 2024 (1): 28- 30.
	LI Q K, LI F, GUO Z. Research on the complexity of air combat based on artificial intelligence[J]. Military Digest, 2024 (1): 28- 30.
7	张易明, 艾剑良. 基于双目视觉的空中加油锥套定位与对接控制[J]. 系统工程与电子技术, 2021, 43 (10): 2940- 2953.
	ZHANG Y M, AI J L. Positioning of aerial refueling drogue and docking control based on binocular vision[J]. Systems Engineering and Electronics, 2021, 43 (10): 2940- 2953.
8	黄舟杭, 张志浩, 韩钧羽, 等. 基于改进遗传算法的远程多点空中加油规划[J]. 火力与指挥控制, 2024, 49 (6): 30- 37.
	HUANG Z H, ZHANG Z H, HUANG J Y, et al. Long-range multi-point air refueling planning based on improved genetic algorithm[J]. Fire Control & Command Control, 2024, 49 (6): 30- 37.
9	郭代言, 宋敏, 侯岳奇, 等. 面向空域巡逻兵力保持的空中加油算法[J]. 兵器装备工程学报, 2024 (S2): 204- 212.
	GUO D Y, SONG M, HOU Y Q, et al. Airspace patrol force maintance oriented air refuelling algorithm[J]. Journal of Ordnance Equipment Engineering, 2024 (S2): 204- 212.
10	李艺辉, 韩旭东. 基于遗传算法的战场空中加油空域规划[J]. 计算机仿真, 2016, 33 (8): 58- 62. doi: 10.3969/j.issn.1006-9348.2016.08.013
	LI Y H, HAN X D. Battlefield air refueling space programming based on genetic algorithm[J]. Computer Simulation, 2016, 33 (8): 58- 62. doi: 10.3969/j.issn.1006-9348.2016.08.013
11	孙勇成, 蔡俊伟, 张帅. 复杂战场环境空中加油空域规划[J]. 指挥信息系统与技术, 2018, 9(3): 59−64.
	SUN Y C, CAI J W, ZHANG S. Airspace deployment for aerial refueling in complex battlefield environment[J]. Command Information System and Technology, 2018, 9(3): 59−64.
12	崔利杰, 祖成昊, 李泽, 等. 基于多种群遗传算法的远程作战体系加油路径优化[J]. 火力与指挥控制, 2017, 42 (6): 88- 92. doi: 10.3969/j.issn.1002-0640.2017.06.020
	CUI L J, ZU C H, LI Z, et al. Route optimization of AAR in long range operational system based on MPGA[J]. Fire Control & Command Control, 2017, 42 (6): 88- 92. doi: 10.3969/j.issn.1002-0640.2017.06.020
13	李晨玉, 邢卓琳, 苏子康, 等. 时空散布下无人机集群空中加油集结路径规划[J]. 无人系统技术, 2024, 7 (5): 13- 23. doi: 10.19942/j.issn.2096-5915.2024.05.44
	LI C Y, XING Z L, SU Z K, et al. Path planning for aerial refueling rendezvous of UAV clusters under spatiotemporal distribution[J]. Unmanned Systems Technology, 2024, 7 (5): 13- 23. doi: 10.19942/j.issn.2096-5915.2024.05.44
14	JIN Z P, SHIMA T, SCHUMACHER C J. Optimal scheduling for refueling multiple autonomous aerial vehicles[J]. IEEE Trans. on Robotics, 2006, 22 (4): 682- 693. doi: 10.1109/TRO.2006.878793
15	MAINI P, SUNDAR K, SINGH M , et al. Cooperative aerial-ground vehicle route planning with fuel constraints for coverage applications[J]. IEEE Trans. on Aerospace and Electronic Systems, 2019, 55(6): 3016−3028.
16	TOYDAS M, MALYEMEZ C. Air refueling optimization for more agile and efficient military deployment operations[J]. The Aeronautical Journal, 2022, 126 (1296): 365- 380. doi: 10.1017/aer.2021.69
17	何旭, 景小宁, 冯超, 等. 基于种群多样性的FPSO算法在空中加油区域配置中的应用[J]. 计算机科学, 2017, 44 (S2): 547- 551.
	HE X, JING X N, FENG C, et al. Diversity-guided FPSO algorithm for solving air refueling region deplaying problem[J]. Computer Science, 2017, 44 (S2): 547- 551.
18	HUANG Z H, ZHANG Z H, HAN J Y, et al. Remote multi-point air refueling planning based on improved differential evolution algorithm[C]//Proc. of the International Conference on Computer Information Science and Application Technology, 2023: 38−44.
19	HANSKNECHT C, JOORMANN I, KORN B, et al. Feeder routing for air-to-air refueling operations[J]. European Journal of Operational Research, 2023, 304 (2): 779- 796. doi: 10.1016/j.ejor.2022.04.016
20	ZHANG Z, HUANG Z, LIU X, et al. Research on multiple air-to-air refueling planning based on multi-dimensional improved NSGA-II algorithm[J]. Electronics, 2023, 12 (18): 3880. doi: 10.3390/electronics12183880
21	HUANG J, XING J, REN J R, et al. Aerial refueling scheduling of multi-receiver and multi-tanker under spatial-temporal constraints for forest firefighting[J]. Chinese Journal of Aeronautics, 2024, 37 (5): 71- 91. doi: 10.1016/j.cja.2024.01.028
22	张西林, 何亚坤, 张恪易, 等. 面向自主空中加油任务的目标检测技术研究[J]. 航空科学技术, 2023, 34 (2): 64- 71. doi: 10.19452/j.issn1007-5453.2023.02.009
	ZHANG X L, HE Y K, ZHANG L Y, et al. Object detection technology for autonomous air to air refueling[J]. Aeronautical Science & Technology, 2023, 34 (2): 64- 71. doi: 10.19452/j.issn1007-5453.2023.02.009
23	XU W L, LUO D L, YANG L, et al. Guidance law research for autonomous aerial refueling of UAV[C]//Proc. of the IEEE International Conference on Cyber Technology in Automation, Control and Intelligent Systems, 2014: 440−445.
24	WANG Y X, CHEN J, WANG H L. Rendezvous path planning of multi-UAV for autonomous aerial refueling[C]//Proc. of the International Conference on Autonomous Unmanned Systems, 2022: 1452−1461.
25	YOON Y, KIM M, KIM Y. Three-dimensional path planning for aerial refueling between one tanker and multiple UAVs[J]. International Journal of Aeronautical and Space Sciences, 2018, 19 (4): 1027- 1040. doi: 10.1007/s42405-018-0098-z
26	钱海力, 秦望龙. 基于聚类方法的多作战任务加油空域规划[J]. 信息化研究, 2020, 46 (3): 17- 22.
	QIAN H L, QIN W L. Multi-mission refueling airspace planning based on clustering methods[J]. Information Research, 2020, 46 (3): 17- 22.
27	TAN R S, GAN X S, WU N, et al. Location method of refueling airspace in air combat based on modified AFS algorithm[C]//Proc. of the IEEE 5th International Conference on Automation: Electronics and Electrical Engineering, 2022: 929−934.
28	FERDOWSI F, MALEKI H R, RIVAZ S. Air refueling tanker allocation based on a multi-objective zero-one integer programming model[J]. Operational Research, 2020, 20, 1913- 1938. doi: 10.1007/s12351-018-0402-5
29	HAO L, QUAN Q. Autonomous aerial refueling of multiple UAVs: an efficient rendezvous scheduling approach[C]//Proc. of the China Automation Congress, 2021: 7463−7468.
30	王连锋, 宋建社, 王正元, 等. 带硬时间窗的战场物资配送车辆路径优化[J]. 系统工程与电子技术, 2013, 35 (4): 770- 776. doi: 10.3969/j.issn.1001-506X.2013.04.15
	WANG L F, SONG J S, WANG Z Y, et al. Vehicle routing optimization with hard time windows in battlefield resources distribution[J]. Systems Engineering and Electronics, 2013, 35 (4): 770- 776. doi: 10.3969/j.issn.1001-506X.2013.04.15
31	张玉玺, 雷冰冰, 王晓峰, 等. 带时间窗的车辆路径问题元启发算法综述[J]. 计算机应用研究, 2025, 42 (5): 1290- 1298.
	ZHANG Y X, LEI B B, WANG X F, et al. Review of metaheuristic algorithms for vehicle routing problem with time windows[J]. Application Research of Computers, 2025, 42 (5): 1290- 1298.
32	MAROOF A, AYVAZ B, NAEEM K. Logistics optimization using hybrid genetic algorithm （HGA）: a solution to the vehicle routing problem with time windows （VRPTW）[J]. IEEE Access, 2024, 12, 36974- 36989. doi: 10.1109/ACCESS.2024.3373699
33	SINTEF. Solom on benchmark problems [EB/OL]. [2024-12-21]. https://www.sintef.no/projectweb/top/vrptw/solomon-benchmark/.
34	WANG H P, HE Y, LI Y, et al. Study on the home health caregiver scheduling problem under a resource sharing mode considering differences in working time and customer satisfaction[J]. Discrete Dynamics in Nature and Society, 2020, 2020 (1): 7360765. doi: 10.1155/2020/7360765
35	LIU S Z, SUN J, DUAN X H, et al. Parallel adaptive large neighborhood search based on spark to solve VRPTW[J]. Scientific Reports, 2024, 14 (1): 23809. doi: 10.1038/s41598-024-74432-2

CAP阵位	坐标/km	威胁等级	时间窗	时段/min	加油需求/kg
1	[0,0]	—	8:00	—	—
2	[10,−400]	低	09:35—09:50	15	9 000
3	[90,−320]	低	09:10—09:25	15	9 000
4	[200,−360]	中	08:30—08:42	12	9 000
5	[270,−250]	中	09:20—09:32	12	9 000
6	[350,−210]	高	08:25—08:35	10	9 000
7	[400,−100]	高	08:50—09:00	10	9 000
8	[430,−20]	高	08:35—08:45	10	9 000
9	[500,60]	中	09:00—09:12	12	9 000
10	[520,140]	中	09:25—09:37	12	9 000
11	[600,180]	中	09:05—09:17	12	9 000
12	[670,200]	中	08:50—09:02	12	9 000
13	[750,300]	低	09:00—09:15	15	9 000

符号	含义	取值
n	时间窗满足度的权重	n =0.05
r	等待时间成本权重	r =0.01
p	时间窗宽度选择权重	p =0.2
q	航段路途成本权重	q =0.2

符号	含义	取值
N_Iteration	ACO算法迭代次数	N_Iteration=50
N_ants	蚁群规模	N_ants=100
ρ	信息素挥发系数	ρ=0.6
α	启发信息权重	α=3
β	信息素浓度权重	β=1

方案	出动架次	等待时间总和	非必需油耗率/%
原始方案	6	78.1	80
优化方案	4	8.93	34.07

解	参数	RC101	RC205
已知最优解	车辆数	15	7
已知最优解	路程	1 619.8	1 157.5
WOA的解^[34]	车辆数	16	7
WOA的解^[34]	路程	1 703.8	1 205.3
ALNS的解^[35]	车辆数	15	7
ALNS的解^[35]	路程	1 656.5	1 337.6
本文最优解	车辆数	15	8
本文最优解	路程	1 707.7	1 199.1
最优解偏离度/%		5.4	3.6