非集装器载重平衡问题建模与两阶段Benders分解启发式算法设计

李云飞; 徐吉辉; 赵向领; 黄激; 童子琛

doi:10.19818/j.cnki.1671-1637.2025.03.019

非集装器载重平衡问题建模与两阶段Benders分解启发式算法设计

doi: 10.19818/j.cnki.1671-1637.2025.03.019

1.
空军工程大学装备管理与无人机工程学院，陕西西安 710051
2.
中国民航大学空中交通管理学院，天津 300300
3.
杭州萧山国际机场有限公司，浙江杭州 311207

基金项目:

国家自然科学基金项目 52272356

国家自然科学基金项目 72461013

详细信息

作者简介:
李云飞(1996-)，男，山西盂县人，空军工程大学博士研究生，从事飞机装载与配平研究

通讯作者:
徐吉辉(1974-)，男，山西闻喜人，空军工程大学教授，博士

中图分类号: U8
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- 被引次数: 0
出版历程
- 收稿日期: 2024-08-23
- 录用日期: 2025-04-30
- 修回日期: 2025-01-30
- 刊出日期: 2025-06-28

Weight balance problem modeling and two-stage Benders decomposition heuristic algorithm design of non-ULDs

1.
Equipment Management and Unmanned Aerial Vehicle Engineering College, Air Force Engineering University, Xi'an 710051, Shaanxi, China
2.
College of Air Traffic Management, Civil Aviation University of China, Tianjin 300300, China
3.
Hangzhou Xiaoshan International Airport Co., Ltd., Hangzhou 311207, Zhejiang, China

Funds:

National Natural Science Foundation of China 52272356

National Natural Science Foundation of China 72461013

More Information

Corresponding author: XU Ji-hui (1974-), male, professor, PhD, skyline6018@126.com

Article Text (Baidu Translation)

摘要

摘要: 为探索民航货机装载非集装器的潜力，研究了货机装载非集装器的载重平衡问题；剖析了非集装器与集装器在载重平衡上的不同，将飞机货舱视为矩形板，非集装器视为矩形块，建立了两阶段的非集装器载重平衡优化模型；在第1阶段的二维几何位置模型中，考虑了非集装器不重叠、不超出货舱边界、可正交旋转等约束，以飞机货舱面积利用率最大为目标函数；在第2阶段的配载模型中，考虑了多种飞机质量和稳定性约束，以装载量最大、重心偏差最小为多目标函数；设计使用了基于逻辑分解的Benders算法，将非集装器的载重平衡问题分解为主问题和子问题；主问题采用改进的遗传模拟算法和最低水平线算法确定非集装器放置顺序和位置，子问题采用y-check算法对各种质量和稳定性等约束检查，并给出了Benders' cut约束模型；设计了非集装器面积大于、小于货舱面积的2种场景，基于本文提出的算法、Gurobi^*、Gurobi和专家配载针对2种不同的装载约束模型进行仿真验证和对比分析。分析结果表明：在货舱左右平衡的二维几何位置分配算例中，Gurobi^*的解质量和求解速度较好，平均装载量、货舱面积利用率、重心偏差、求解时间分别为19 872 kg、65.88%、2.08%MAC、61.18 s；专家配载结果相对较差，平均装载量、货舱面积利用率、重心偏差、求解时间分别为18 494 kg、65.21%、2.79% MAC、986.98 s；提出的算法作为一种启发式方法，平均装载量为18 874 kg，略低于Gurobi^*和Gurobi的优化结果，但平均货舱面积利用率和重心偏差分别为71.87%、2.76%MAC，且平均求解速度为175.97 s，明显快于Gurobi的1 082.92 s。建立的两阶段载重平衡优化模型和算法能够为非集装器装载位置和方向的确定提供参考。
- 航空运输 /
- 非集装器 /
- Benders分解 /
- 载重平衡问题 /
- 多目标优化 /
- 二维切割
Abstract: To explore the potential of loading non-unit load devices (non-ULDs) in civil aviation cargo aircraft, the weight balance problem (WBP) of loading non-ULDs in cargo aircraft was studied. The differences between loading non-ULDs and ULDs in WBP were compared. The cargo hold was regarded as a rectangular plane and the non-ULDs as rectangular items, a two-stage weight balance optimization model was constructed for loading non-ULDs. In the two-dimensional geometric constraint model in the first stage, constraints of non-ULDs were considered such as no overlapping, not exceeding the cargo hold boundaries, and orthogonal rotation. The maximum utilization of the aircraft cargo hold plane area was taken as the objective function. In the weight balance model in the second stage, constraints of various aircraft were considered such as weight and stability. The maximum payload and minimum center of gravity (CG) deviation were selected as the multi-objective function. The Benders algorithm was designed and used based on logical decomposition. The WBP of cargo aircraft loading non-ULDs was decomposed into a master problem and a sub-problem. The master problem used improved genetic simulation and the lowest horizontal line algorithm to determine the loading sequence and position of non-ULDs. The sub-problem employed the y-check algorithm to check various constraints, such as weight and stability. The Benders' cut constraint model was provided. The two scenarios where the area of non-ULDs was greater than and less than the cargo hold area were designed. The model was verified and compared using four methods: the proposed algorithm, Gurobi^*, Gurobi, and expert stowing for two different loading constraint requirements. Experimental results show that, when testing a two-dimensional geometric position allocation model for left-right balance in the cargo hold, Gurobi^* achieves the best solution quality and speed. The average payload, cargo hold area utilization rate, CG deviation, and solution time are 19 872 kg, 65.88%, 2.08%MAC and 61.18 s, respectively. The expert stowing method is the worst, with its average payload, cargo hold area utilization rate, CG deviation, and solution time reaching 18 494 kg, 65.21%, 2.79%MAC, and 986.98 s, respectively. As a heuristic algorithm, the proposed algorithm gets an average payload of 18 874 kg, slightly worse than the optimized solutions of Gurobi^* and Gurobi. The average cargo hold area utilization rate and CG deviation are 71.87% and 2.76%MAC, respectively. The average solution time is 175.97 s, much faster than Gurobi's of 1 082.92 s. The proposed two-stage weight balance optimization model and algorithm can provide a reference for determining the loading positions and directions of non-ULDs.
- air transport /
- non-ULD /
- Benders decomposition /
- weight balance problem /
- multi-objective optimization /
- two-dimensional cutting

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图 1 标准集装器与非集装器的区别

Figure 1. Difference between standard and non-ULDs

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图 2 飞机货舱平面坐标系

Figure 2. Plane coordinate system of aircraft cargo hold

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图 3 各货舱质量限制(单位: kg)

Figure 3. Weight limits for each cargo hold (unit: kg)

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图 4 货舱h质量分配

Figure 4. Weight distribution of cargo hold h

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图 5 最低水平线算法

Figure 5. Lowest horizontal line algorithm

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图 6 编码方式

Figure 6. Coding method

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图 7 交叉方式

Figure 7. Crossing method

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图 8 变异方式

Figure 8. Variation method

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图 9 遗传模拟退火算法

Figure 9. Genetic simulated annealing algorithm

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图 10 模型1装载量对比

Figure 10. Comparison of payloads of model 1

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图 11 模型1货舱面积利用率对比

Figure 11. Comparison of cargo hold area utilization rates of model 1

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图 12 模型1重心偏差对比

Figure 12. Comparison of CG deviations of model 1

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图 13 模型1求解时间对比

Figure 13. Comparison of solution times of model 1

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图 14 算例2-9正交旋转装载结果

Figure 14. Orthogonal rotation loading results of test 2-9

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图 15 算例2-9不可正交旋转装载结果

Figure 15. Non-orthogonal rotation loading results of test 2-9

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图 16 模型2装载量对比

Figure 16. Comparison of payloads of model 2

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图 17 模型2货舱面积利用率对比

Figure 17. Comparison of cargo hold area utilization rates of model 2

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图 18 模型2重心偏差对比

Figure 18. Comparison of CG deviations of model 2

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图 19 模型2求解时间对比

Figure 19. Comparison of solution times of model 2

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图 20 算例1-3正交旋转装载结果

Figure 20. Orthogonal rotation loading results of test 1-3

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图 21 算例1-3不可正交旋转装载结果

Figure 21. Non-orthogonal rotation loading results of test 1-3

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图 22 Gurobi装载结果

Figure 22. Loading results of Gurobi

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图 23 Gurobi^*装载结果

Figure 23. Loading results of Gurobi^*

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表 1 非集装器数据

Table 1. Non-ULD data

编号	长/m	宽/m	质量/kg	编号	长/m	宽/m	质量/kg
1	3.00	2.44	4 500	16	4.22	2.11	1 090
2	6.88	1.87	4 440	17	4.22	2.11	1 073
3	6.00	2.44	4 077	18	3.73	2.32	770
4	5.69	2.04	3 472	19	3.68	1.88	670
5	5.61	2.13	3 313	20	4.04	1.98	670
6	5.61	2.04	3 296	21	6.11	3.41	3 532
7	5.69	2.01	3 270	22	5.92	3.69	2 056
8	5.61	2.01	3 270	23	5.39	2.12	1 668
9	5.61	2.13	3 270	24	5.21	3.60	3 931
10	5.61	2.04	3 270	25	5.94	2.34	3 477
11	5.69	2.03	3 200	26	7.32	3.46	5 368
12	2.74	2.24	2 850	27	7.18	3.24	5 456
13	4.11	2.11	2 100	28	5.30	3.26	4 599
14	4.41	2.41	1 845	29	6.73	3.24	3 358
15	4.14	2.11	1 700	30	6.17	3.01	2 687

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表 2 飞机基本参数

Table 2. Basic parameters of aircraft

参数	数值	参数	数值
I_O	31.3	R	70 000
P_S/%MAC	23	D/m	26.36
Q_O/kg	52 752	B_L/m	25.19
Q_M/kg	30 708	B_M/m	5.07
I_{T, F}	4.5	L/m	47.3
I_{L, F}	0.4	W/m	3.7

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表 3 模型1装载量

Table 3. Payloads of model 1

类型	数量	算例	装载量/kg
类型	数量	算例	本文算法	Gurobi^*	Gurobi	专家
1	20	1-1	27 185	27 994	27 994	26 678
		1-2	26 235	27 388	27 388	25 112
		1-3	28 220	30 379	30 379	27 543
	10	1-4	27 493	28 819	28 819	26 362
		1-5	24 305	20 189	20 189	21 876
		1-6	23 832	20 723	20 723	19 109
		1-7	26 750	28 481	28 481	24 098
	5	1-8	10 735	12 998	12 998	9 798
		1-9	6 965	7 010	7 010	10 032
		1-10	4 286	2 065	2 065	3 298
2	20	2-1	30 181	30 573	30 573	29 177
		2-2	29 639	30 414	30 414	27 843
		2-3	30 115	30 648	30 648	28 754
	10	2-4	23 628	23 222	23 222	19 803
		2-5	12 359	11 376	11 376	10 873
		2-6	20 051	21 410	21 410	22 892
		2-7	14 227	15 859	15 859	12 565
	5	2-8	6 617	5 283	5 283	4 695
		2-9	18 238	18 238	18 238	15 987
		2-10	8 584	8 584	8 584	6 003
均值			19 982	20 083	20 083	18 625

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表 4 模型1货舱面积利用率

Table 4. Cargo hold area utilization rates of model 1

类型	数量	算例	货舱面积利用率/%
类型	数量	算例	本文算法	Gurobi^*	Gurobi	专家
1	20	1-1	76.46	80.53	80.53	68.33
		1-2	75.88	74.51	74.51	77.98
		1-3	58.19	70.77	70.77	64.09
	10	1-4	82.15	67.15	67.15	79.03
		1-5	65.45	76.57	76.57	67.09
		1-6	70.62	78.40	78.40	66.76
		1-7	89.90	79.21	79.21	73.09
	5	1-8	79.09	76.84	76.84	67.89
		1-9	79.14	40.00	40.00	54.89
		1-10	60.72	17.52	17.52	43.12
2	20	2-1	63.49	70.26	70.26	67.32
		2-2	64.52	77.00	77.00	74.01
		2-3	69.85	72.45	72.45	68.32
	10	2-4	74.63	83.20	83.20	75.07
		2-5	72.45	57.91	57.91	64.06
		2-6	84.02	79.59	79.59	72.98
		2-7	71.43	63.99	63.99	58.23
	5	2-8	48.28	46.79	46.79	40.77
		2-9	73.24	73.24	73.24	73.24
		2-10	72.99	72.99	72.99	72.99
均值			71.62	67.95	67.95	66.46

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表 5 模型1重心偏差

Table 5. CG deviations of model 1

类型	数量	算例	重心偏差/%MAC
类型	数量	算例	本文算法	Gurobi^*	Gurobi	专家
1	20	1-1	1.99	1.78	1.07	2.32
		1-2	2.35	1.89	1.46	2.65
		1-3	2.01	1.36	1.89	2.95
	10	1-4	2.36	1.83	1.51	2.66
		1-5	1.96	0.65	1.24	2.39
		1-6	1.36	0.53	1.84	1.99
		1-7	1.69	0.96	0.62	1.85
	5	1-8	0.97	1.86	1.29	2.31
		1-9	2.32	2.45	2.94	2.88
		1-10	1.67	2.77	3.97	2.93
2	20	2-1	2.56	1.72	1.37	2.97
		2-2	2.31	1.45	1.36	3.11
		2-3	1.96	0.50	0.43	2.75
	10	2-4	1.32	1.59	0.69	2.39
		2-5	1.66	2.35	1.48	1.85
		2-6	1.25	1.87	1.70	1.69
		2-7	2.31	0.31	1.63	2.63
	5	2-8	1.98	3.12	2.97	2.85
		2-9	1.66	1.51	1.14	1.76
		2-10	0.99	0.68	1.34	1.99
均值			1.83	1.56	1.60	2.45

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表 6 模型1求解时间

Table 6. Solution times of model 1

类型	数量	算例	时间/s
类型	数量	算例	本文算法	Gurobi^*	Gurobi	专家
1	20	1-1	634.74	466.02	3 376.02	1 932.36
		1-2	656.11	256.75	3 601.94	1 853.44
		1-3	495.68	254.05	3 601.39	1 736.66
	10	1-4	84.32	2.56	4.13	1 803.65
		1-5	60.74	3.65	7.32	1 855.32
		1-6	71.66	2.68	4.22	1 400.93
		1-7	56.98	1.63	3.33	1 362.57
	5	1-8	20.14	1.06	0.35	1 001.44
		1-9	24.96	1.23	0.78	957.32
		1-10	20.12	0.67	0.11	838.53
2	20	2-1	556.65	1.67	2.07	2 100.65
		2-2	514.20	230.45	3 600.84	1 988.32
		2-3	537.78	357.36	3 601.44	2 045.66
	10	2-4	95.31	3.43	5.71	1 700.63
		2-5	84.69	0.78	0.44	1 756.92
		2-6	74.87	19.69	205.05	1 823.99
		2-7	63.98	0.39	0.81	1 423.55
	5	2-8	32.01	1.98	0.43	639.75
		2-9	19.36	0.69	0.94	800.54
		2-10	22.54	0.67	0.86	755.44
均值			206.34	80.37	900.91	1 488.89

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表 7 模型2装载量

Table 7. Payloads of model 2

类型	数量	算例	装载量/kg
类型	数量	算例	本文算法	Gurobi^*	Gurobi	专家
1	20	1-1	22 135	23 924	23 924	25 536
		1-2	27 235	28 740	28 740	24 262
		1-3	29 022	30 697	30 697	28 635
	10	1-4	27 493	28 819	28 819	26 543
		1-5	21 208	22 532	22 532	18 552
		1-6	22 412	22 365	22 365	18 435
		1-7	27 452	29 874	29 874	26 792
	5	1-8	10 078	11 963	11 963	9 654
		1-9	11 189	8 563	8 563	6 065
		1-10	4 630	3 560	3 560	4 120
2	20	2-1	28 742	30 022	30 022	28 541
		2-2	20 023	22 434	22 434	28 455
		2-3	26 354	29 977	29 977	28 436
	10	2-4	23 301	23 222	23 222	20 013
		2-5	10 014	11 379	11 379	9 325
		2-6	19 783	21 410	21 410	18 652
		2-7	13 620	15 859	15 859	13 204
	5	2-8	5 961	5 283	5 283	7 836
		2-9	18 238	18 238	18 238	18 238
		2-10	8 584	8 584	8 584	8 584
均值			18 874	19 872	19 872	18 494

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表 8 模型2货舱面积利用率

Table 8. Cargo hold area utilization rates of model 2

类型	数量	算例	货舱面积利用率/%
类型	数量	算例	本文算法	Gurobi^*	Gurobi	专家
1	20	1-1	75.63	80.00	80.00	70.64
		1-2	72.63	74.75	74.75	67.15
		1-3	65.32	67.07	67.07	66.35
	10	1-4	72.36	67.15	67.15	63.26
		1-5	63.25	70.63	70.63	60.32
		1-6	71.36	72.54	72.54	68.58
		1-7	83.45	74.34	74.34	75.65
	5	1-8	64.98	70.71	70.71	74.55
		1-9	81.58	42.63	42.63	37.32
		1-10	69.77	25.75	25.75	58.64
2	20	2-1	73.69	69.35	69.35	58.95
		2-2	64.88	58.91	58.91	60.78
		2-3	60.16	66.11	66.11	65.94
	10	2-4	76.58	83.20	83.20	75.12
		2-5	77.66	57.91	57.91	66.91
		2-6	83.46	79.59	79.59	74.32
		2-7	74.64	63.99	63.99	60.88
	5	2-8	59.78	46.79	46.79	52.65
		2-9	73.24	73.24	73.24	73.24
		2-10	72.99	72.99	72.99	72.99
均值			71.87	65.88	65.88	65.21

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表 9 模型2重心偏差

Table 9. CG deviations of model 2

类型	数量	算例	重心偏差/%MAC
类型	数量	算例	本文算法	Gurobi^*	Gurobi	专家
1	20	1-1	2.01	3.99	1.07	2.66
		1-2	2.77	1.53	1.46	3.51
		1-3	2.64	1.89	1.89	2.77
	10	1-4	2.02	1.86	1.51	2.44
		1-5	2.77	1.36	1.58	2.57
		1-6	3.72	1.87	2.21	2.01
		1-7	1.69	1.23	1.75	2.47
	5	1-8	3.70	0.89	1.32	2.78
		1-9	1.51	2.36	1.85	2.97
		1-10	3.64	1.85	1.27	3.54
2	20	2-1	2.99	1.87	1.34	3.65
		2-2	3.73	3.95	3.25	2.78
		2-3	3.62	1.98	2.78	2.96
	10	2-4	3.37	1.23	1.45	2.53
		2-5	2.97	3.45	2.97	2.85
		2-6	3.86	1.34	1.66	2.99
		2-7	3.72	2.03	1.65	2.52
	5	2-8	1.45	2.01	2.34	3.63
		2-9	1.82	1.93	1.07	2.35
		2-10	1.12	2.93	2.85	1.87
均值			2.76	2.08	1.86	2.79

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表 10 模型2求解时间

Table 10. Solution times of model 2

类型	数量	算例	时间/s
类型	数量	算例	本文算法	Gurobi^*	Gurobi	专家
1	20	1-1	736.56	200.02	3 602.61	1 935.36
		1-2	591.44	107.42	3 601.94	1 960.79
		1-3	391.83	478.54	3 601.59	1 803.41
	10	1-4	88.45	3.60	1.20	1 432.84
		1-5	74.21	1.47	3.61	823.16
		1-6	60.92	3.90	3.74	635.53
		1-7	53.29	2.56	3.12	721.54
	5	1-8	43.64	1.65	2.00	536.99
		1-9	14.83	2.06	2.71	588.47
		1-10	40.08	2.69	2.50	698.12
2	20	2-1	347.20	134.86	3 604.73	765.64
		2-2	636.04	18.44	3 601.30	2 003.69
		2-3	107.90	96.71	3 601.71	2 108.74
	10	2-4	103.49	6.13	3.38	753.51
		2-5	44.14	0.06	1.76	798.35
		2-6	14.80	161.81	11.66	678.35
		2-7	38.35	0.16	1.97	699.42
	5	2-8	31.88	0.84	2.43	206.58
		2-9	27.24	0.31	2.89	325.78
		2-10	73.15	0.42	1.55	263.41
均值			175.97	61.18	1 082.91	986.98

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