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基于装卸顺序的中型机多航段协同配载优化

赵向领 李云飞 王治宇 徐吉辉 李鹏飞

赵向领,李云飞,王治宇,等. 基于装卸顺序的中型机多航段协同配载优化[J]. 北京航空航天大学学报,2024,50(4):1147-1161 doi: 10.13700/j.bh.1001-5965.2022.0439
引用本文: 赵向领,李云飞,王治宇,等. 基于装卸顺序的中型机多航段协同配载优化[J]. 北京航空航天大学学报,2024,50(4):1147-1161 doi: 10.13700/j.bh.1001-5965.2022.0439
ZHAO X L,LI Y F,WANG Z Y,et al. Cooperating loading balance optimization for medium-sized aircraft with multiple flight legs based on loading and unloading sequence[J]. Journal of Beijing University of Aeronautics and Astronautics,2024,50(4):1147-1161 (in Chinese) doi: 10.13700/j.bh.1001-5965.2022.0439
Citation: ZHAO X L,LI Y F,WANG Z Y,et al. Cooperating loading balance optimization for medium-sized aircraft with multiple flight legs based on loading and unloading sequence[J]. Journal of Beijing University of Aeronautics and Astronautics,2024,50(4):1147-1161 (in Chinese) doi: 10.13700/j.bh.1001-5965.2022.0439

基于装卸顺序的中型机多航段协同配载优化

doi: 10.13700/j.bh.1001-5965.2022.0439
基金项目: 国家自然科学基金(71802141,52074309,U2333217);南航载重平衡管理系统提升(KC22204);中央高校基本科研业务费专项资金(3122020049);中国民航大学研究生科研创新资助项目(2021YJS060)
详细信息
    通讯作者:

    E-mail:zxl-llx@163.com

  • 中图分类号: U8;V271.9;O221

Cooperating loading balance optimization for medium-sized aircraft with multiple flight legs based on loading and unloading sequence

Funds: National Natural Science Foundation of China (71802141,52074309,U2333217); Improvement of Loading Balance Management System for China Southern Airlines (KC22204); The Fundamental Research Funds for the Central Universities (3122020049); Graduate Research and Innovation Funding Project of Civil Aviation University of China (2021YJS060)
More Information
  • 摘要:

    多航段、多经停的航空货运航班,飞行过程中需要合理控制飞机的重心(CG)位置,保持平衡状态,中转站装卸操作过程中需要避免额外装卸操作,减少时间和劳力浪费,因此,研究多航段航班的协同配载问题具有重要现实意义。根据航空器自身特点与集装器(ULD)装卸顺序,结合各航段飞机装载平衡与过站机场ULD装卸操作要求,通过协调分配各航段ULD和散货位置,建立了中型机联程航班多目标整数线性规划模型。模型考虑了ULD和散货的质量、体积及与舱位的匹配限制,航空器限重、舱位限重、舱位及区域累积限重、上下舱联合限重和CG位置的平衡限制,以及前后航段ULD和散货的连接性约束。根据造成中间机场额外装卸操作的2种原因,基于装卸顺序,提出装卸优化模型,通过ULD的舱内平移,优化了CG,减少了装卸次数。以B757-200F机型为例,在两装一卸、一装两卸和两装两卸3种场景下,采用商用求解器Gurobi,分别针对3种不同目标函数组合进行求解和分析对比。实验表明:所提模型可以有效协调过站货物的额外装卸次数,优化前后两航段的CG位置。

     

  • 图 1  飞机转运示意图

    Figure 1.  Schematic diagram of aircraft transfer

    图 2  B757-200F货舱侧视图

    Figure 2.  Side view of B757-200F cargo compartment

    图 3  各航段起飞重心偏差雷达图

    Figure 3.  Radar chart of take-off CG deviation in each leg

    图 4  中间机场额外装卸次数

    Figure 4.  Number of extra operations at intermediate airports

    图 5  模型求解时间

    Figure 5.  Model solution time

    图 6  算例1-1综合优化配载结果

    Figure 6.  Test 1-1 comprehensive optimization stowage results

    图 7  算例1-1偏好装卸配载结果

    Figure 7.  Test 1-1 preferred loading and unloading stowage results

    图 8  算例1-1偏好重心配载结果

    Figure 8.  Test 1-1 preferred center of gravity stowage results

    图 9  算例2-3综合优化配载结果

    Figure 9.  Test 2-3 comprehensive optimization stowage results

    图 10  算例2-3偏好装卸配载结果

    Figure 10.  Test 2-3 preferred loading and unloading stowage results

    图 11  算例2-3偏好重心配载结果

    Figure 11.  Test 2-3 preferred center of gravity stowage results

    图 12  算例3-8综合优化配载结果

    图 13  算例3-8偏好装卸配载结果

    Figure 13.  Test 3-8 preferred loading and unloading stowage results

    图 14  算例3-8偏好重心配载结果

    Figure 14.  Test 3-8 preferred center of gravity stowage results

    表  1  上下舱联合区间限重

    Table  1.   Combined weight limit of upper and lower cabins

    区间 实际区间联合载量计算公式 区间限重/kg
    CZL CL1 C15 2716
    CL2 CL1+C2+D1×10% 4360
    CL3 CL2+C1+D1×80% 7871
    CL4 CL3+C3+D1×10%+D2×40% 11557
    CL5 CL4+C4+D2×40% 14043
    CL6 CL5+C5+D2×20% 15893
    CL7 CL6+C6 17712
    CZR CR8 CR7+C7 23748
    CR7 CR6+C8 21909
    CR6 CR5+C9+D3×20% 20102
    CR5 CR4+C10+D3×50% 18257
    CR4 CR3+C11+D3×30%+D4×10% 15733
    CR3 CR2+C12+D4×40% 11969
    CR2 CR1+C13+D4×30% 8239
    CR1 C14+D4×20% 3476
    下载: 导出CSV

    表  2  B757-200F基本参数

    Table  2.   Basic parameters of B757-200F

    WOEW/kg WMTOW/kg WMZFW/kg BMAC/m CD/m IOEW WMLDW/kg PCGT,l /MAC BLEMAC/m ε
    52752 108862 83460 5.07 26.36 31.3 95254 23 25.19 70000
    下载: 导出CSV

    表  3  主货舱舱位约束

    Table  3.   Main cargo space constraints

    舱位 力臂/m 最大载荷/kg 最大高度/m 分区载荷
    C15 9.91 2716 2 C15C2C1C3
    C4质量和
    不超过18000 kg
    C2 12.17 2948 2
    C1 14.43 2948 2
    C3 16.69 2948 2
    C4 18.95 2948 2
    C5 21.21 2948 2 C5C9质量和
    不超过24000 kg
    C6 23.47 2948 2
    C7 25.73 4264 2
    C8 27.99 4264 2
    C9 30.25 2948 2
    C10 32.51 2948 2 C10C14质量和
    不超过29000 kg
    C11 34.77 2948 2
    C12 37.03 2948 2
    C13 29.29 2948 2
    C14 41.55 2948 1.95
    下载: 导出CSV

    表  4  下货舱舱位约束

    Table  4.   Lower cargo hold space constraints

    下货舱舱位 舱位限重/kg 舱位累积限重/kg 力臂/m 可用体积/m3
    D1 2496 4672 13.58 4.9
    D2 4672 4672 18.64 13.9
    D3 3773 7393 32.73 14.2
    D4 5606 7393 38.29 16.7
    下载: 导出CSV

    表  5  油量指数

    Table  5.   Fuel index

    F/kg I F/kg I F/kg I
    500 +0.1 1000 +0.2 1500 +0.3
    2000 +0.4 2500 +0.5 3000 +0.7
    3500 +0.8 4000 +1.0 4500 +1.2
    5000 +1.4 5500 +1.6 6000 +1.8
    7000 +2.3 7500 +2.6 8000 +2.9
    8500 +3.2 9000 +3.6 9500 +4.0
    10000 +4.5 10500 +5.0 11000 +5.6
    11500 +6.2 12500 +7.6 15000 +7.5
    15500 +7.1 16000 +6.7 17000 +5.9
    18000 +5.1 19000 +4.3 20000 +3.5
    24000 +0.6 26000 −0.8 26500 −1.2
    27000 −1.6 27500 −1.9
    下载: 导出CSV

    表  6  货物装载清单

    Table  6.   Cargo loading list

    测试
    场景
    ULD
    规模
    算例
    序号
    U1/
    (kg(个))
    U2/
    (kg(个))
    U3/
    (kg(个))
    B1/
    (kg(个))
    B2/
    (kg(个))
    B3/
    (kg(个))
    WTOF,1/
    kg
    WRFW,1/
    kg
    WMPL,1/
    kg
    WTOF,2/
    kg
    WRFW,2/
    kg
    WMPL,2/
    kg
    两装
    一卸
    < 1-1 0 4634(3) 15092(10) 0 763(50) 2221(150) 7000 500 30708 15000 2500 30708
    1-2 0 6054(5) 13971(9) 0 1028(70) 1996(140) 8000 1000 30708 16000 2000 30708
    1-3 0 11271(7) 9599(6) 0 1355(90) 1184(80) 7500 1000 30708 15500 2500 30708
    = 1-4 0 15618(10) 6848(5) 0 2361(150) 1521(100) 8500 1500 30708 16000 2000 30708
    1-5 0 18655(12) 4588(3) 0 2752(180) 2353(150) 8000 1000 30708 17000 3000 30708
    1-6 0 14071(9) 7952(6) 0 2400(160) 3078(200) 9000 1000 30708 15500 2000 30708
    1-7 0 4568(4) 18306(11) 0 3080(200) 3629(240) 9000 1500 30708 24000 4000 30708
    > 1-8 0 30540(20(10)) 7220(5) 0 2166(150) 1635(100) 8500 1000 30708 26500 5000 29610
    1-9 0 25170(18(6)) 13178(9) 0 1545(100) 2043(130) 9500 2000 30708 27500 5500 28610
    1-10 0 10869(7) 13134(10(8)) 0 1513(100) 2324(150) 8000 500 30708 26000 5000 30110
    一装
    两卸
    < 2-1 2672(2) 0 18155(11) 1398(80) 0 2801(160) 16000 3000 30708 7000 500 30708
    2-2 6111(4) 0 11424(8) 2062(120) 0 2604(150) 15000 2500 30708 7500 1000 30708
    2-3 13034(9) 0 6604(4) 2617(150) 0 2100(120) 15500 2000 30708 8000 1000 30708
    = 2-4 10438(6) 0 14426(9) 1685(100) 0 2665(150) 17000 3000 30708 8500 1500 30708
    2-5 11036(8) 0 9902(7) 2275(130) 0 2023(110) 26500 5000 29610 9000 2000 30708
    2-6 11742(9) 0 8548(6) 2560(150) 0 1824(100) 27500 6000 28610 9500 2000 30708
    2-7 17643(11) 0 6349(4) 2891(160) 0 2118(120) 27000 5500 29110 9500 1000 30708
    > 2-8 29946(20(9)) 0 10087(6) 3540(200) 0 1757(100) 18000 3500 30708 10000 2000 30708
    2-9 22490(15(8)) 0 10603(7) 2773(160) 0 2144(120) 19000 4000 30708 11000 2000 30708
    2-10 10863(7) 0 18568(13(8)) 2118(120) 0 2522(140) 20000 4500 30708 11500 2000 30708
    两装
    两卸
    < 3-1 2587(2) 8018(5) 15230(10) 60/947 1499(100) 1729(120) 10500 2500 30708 10500 3000 30708
    3-2 11246(7) 11627(8) 10205(7) 1834(120) 2330(140) 1757(120) 11000 3000 30708 9500 1500 30708
    3-3 12059(8) 7074(5) 10593(7) 2023(140) 1490(100) 1804(120) 10500 3000 30708 10000 2000 30708
    = 3-4 3078(2) 2836(2) 21057(13) 1513(60) 2271(80) 4242(160) 9500 1500 30708 11000 3000 30708
    3-5 8074(5) 6568(5) 14278(10) 1851(100) 1587(100) 1711(100) 10000 2000 30708 10000 2000 30708
    3-6 13753(9) 10177(6) 9400(6) 4415(150) 3032(110) 2964(110) 11000 3000 30708 12500 2500 30708
    3-7 19430(12) 16475(12) 3827(3) 5059(180) 5028(180) 2242(80) 10500 3000 30708 9500 1500 30708
    > 3-8 16679(10(6)) 16320(10(6)) 13056(9) 2830(160) 2888(160) 1749(100) 12500 2500 30708 10000 2000 30708
    3-9 7563(5) 11576(8(5)) 23684(15(10)) 2778(100) 2765(100) 2844(100) 9500 1500 30708 10000 2000 30708
    3-10 10022(7) 12273(9(7)) 17413(13(8)) 2732(120) 2689(120) 3102(140) 10000 2000 30708 11000 3000 30708
     注:表中数据形式Z(X(Y))和Z(X)中,X表示各场景下待装ULD或散货数量,Z表示待装质量,Y为当前航空器可装载ULD数量,仅当X>Y时表示为X(Y)。
    下载: 导出CSV

    表  7  各航段业载分配

    Table  7.   Payload distribution of each leg

    算例
    序号
    U/个 B/件 WPL,l /kg
    l=1 l=2 l=1 l=2 l=1 l=2
    1-1 10 13 150 220 17313 22710
    1-2 9 14 140 210 15967 23049
    1-3 6 13 80 170 10783 23389
    1-4 5 15 100 250 8369 26348
    1-5 3 15 150 330 6941 28348
    1-6 6 15 200 360 11030 27501
    1-7 11 15 240 440 21935 29583
    1-8 5 15 100 250 8855 28217
    1-9 9 15 130 230 15221 27231
    1-10 8 15 150 250 13303 25685
    2-1 13 11 240 160 25026 20956
    2-2 12 8 270 150 22201 14028
    2-3 13 4 270 120 24355 8704
    2-4 15 9 250 150 29214 17091
    2-5 15 7 240 110 25236 11925
    2-6 15 6 250 100 24674 10372
    2-7 15 4 280 120 29001 8467
    2-8 15 6 300 100 30704 11844
    2-9 15 7 280 120 29199 12747
    2-10 15 8 260 140 28638 15657
    3-1 12 15 180 220 20493 26476
    3-2 14 15 240 260 25042 25919
    3-3 15 12 260 220 26479 20961
    3-4 15 15 220 240 29890 30406
    3-5 15 15 200 200 25914 24144
    3-6 15 15 260 220 30442 25573
    3-7 15 15 260 260 30558 27572
    3-8 15 15 260 260 28637 29910
    3-9 15 15 200 200 30312 30703
    3-10 15 15 260 260 27153 27103
    下载: 导出CSV

    表  8  重心偏差

    Table  8.   Center of gravity deviation

    算例
    序号
    CMAC,l|/%MAC
    (综合优化)
    CMAC,l|/%MAC
    (偏好装卸)
    CMAC,l|/%MAC
    (偏好重心)
    l=1 l=2 l=1 l=2 l=1 l=2
    1-1 0.894 0.759 4.960 7.336 0.439 0.372
    1-2 0.898 0.748 2.165 0.282 0.441 0.366
    1-3 0.394 0.745 0.843 1.741 0.496 0.324
    1-4 0.512 0.708 0.079 0.852 0.579 0.663
    1-5 1.538 0.878 2.658 1.090 1.538 1.056
    1-6 0.975 0.404 10.176 8.362 0.480 0.351
    1-7 1.652 1.271 7.025 3.508 0.403 0.314
    1-8 0.312 0.663 3.927 1.517 0.484 0.311
    1-9 0.889 0.637 9.953 5.567 0.437 0.311
    1-10 0.028 0.517 5.711 3.374 0.458 0.320
    2-1 0.514 0.832 4.045 3.528 0.398 0.375
    2-2 0.146 0.897 1.665 4.501 0.410 0.413
    2-3 0.363 0.489 2.986 4.616 0.773 0.994
    2-4 0.370 0.910 3.557 2.669 0.339 0.432
    2-5 0.183 0.491 3.370 0.193 0.655 0.936
    2-6 0.210 0.949 6.829 0.700 0.602 0.135
    2-7 0.629 0.976 4.772 8.388 0.581 0.453
    2-8 0.675 0.924 3.432 4.481 0.644 0.911
    2-9 0.011 0.795 3.701 2.530 0.332 0.443
    2-10 0.450 0.850 3.975 2.854 0.607 0.853
    3-1 0.852 0.388 2.502 1.894 0.403 0.375
    3-2 0.811 0.805 3.963 3.991 0.379 0.382
    3-3 0.766 0.822 3.922 2.374 0.375 0.403
    3-4 0.396 0.388 4.117 0.671 0.364 0.357
    3-5 0.802 0.791 3.968 4.049 0.380 0.388
    3-6 0.198 0.631 3.111 0.708 0.357 0.370
    3-7 0.358 0.375 1.619 2.887 0.358 0.375
    3-8 0.737 0.741 7.375 5.180 0.358 0.363
    3-9 0.363 0.360 0.773 1.454 0.363 0.360
    3-10 0.179 0.055 4.302 2.786 0.179 0.054
    下载: 导出CSV

    表  9  额外装卸次数

    Table  9.   Number of extra operations

    算例
    序号
    Eunload 时间/s
    综合优化 偏好装卸 偏好重心 综合优化 偏好装卸 偏好重心
    1-1 0 0 10 8.39 0.48 8.46
    1-2 9 8 9 0.39 0.45 5.21
    1-3 2 6 6 0.33 0.25 0.12
    1-4 4 4 5 0.28 0.22 0.19
    1-5 0 0.27 0 3 0.45 0.19
    1-6 0 0 6 4.05 0.64 10.75
    1-7 0 0 11 1.57 0.63 9.48
    1-8 3 1 5 0.58 0.5 0.34
    1-9 0 0 9 7.98 0.46 8.07
    1-10 8 8 9 1.27 0.91 3.46
    2-1 10 11 11 0.63 0.56 3.98
    2-2 4 7 7 0.41 0.41 3.72
    2-3 0 0 4 0.57 0.29 2.79
    2-4 8 8 9 0.58 0.62 5.02
    2-5 6 7 7 0.47 0.44 1.61
    2-6 6 5 6 0.27 0.23 0.08
    2-7 4 4 4 0.34 0.2 0.12
    2-8 6 4 6 1.19 0.37 0.1
    2-9 6 6 7 0.35 0.21 7.81
    2-10 7 8 8 100.75 0.94 6.69
    3-1 1 1 10 1.32 0.74 3.87
    3-2 7 7 7 5.05 2.45 1.73
    3-3 7 7 7 2.79 1 15.53
    3-4 0 0 13 0.94 0.81 3.4
    3-5 1 0 10 9.17 0.69 13.44
    3-6 6 6 6 0.44 3.04 1.78
    3-7 0 0 3 10.4 2.4 8.05
    3-8 0 0 9 7.81 1.48 5.9
    3-9 0 0 10 76.79 1.97 38.45
    3-10 0 0 8 7.87 2.03 3.94
    下载: 导出CSV
  • [1] FENG B, LI Y Z, SHEN Z J M. Air cargo operations: Literature review and comparison with practices[J]. Transportation Research Part C:Emerging Technologies, 2015, 56: 263-280. doi: 10.1016/j.trc.2015.03.028
    [2] AMIOUNY S V, BARTHOLDI J J, VANDE VATE J H, et al. Balanced loading[J]. Operations Research, 1992, 40(2): 238-246. doi: 10.1287/opre.40.2.238
    [3] WODZIAK J R, FADEL G M. Packing and optimizing the center of gravity location using a genetic algorithm[J]. Journal of Computers in Industry, 1994, 11: 2-14.
    [4] MATHUR K. An integer-programming-based heuristic for the balanced loading problem[J]. Operations Research Letters, 1998, 22(1): 19-25. doi: 10.1016/S0167-6377(97)00044-8
    [5] HEIDELBERG K R, PARNELL G S, AMES J E. Automated air load planning[J]. Naval Research Logistics, 1998, 45(8): 751-768. doi: 10.1002/(SICI)1520-6750(199812)45:8<751::AID-NAV1>3.0.CO;2-R
    [6] DAHMANI N, KRICHEN S. On solving the bi-objective aircraft cargo loading problem[C]//Proceedings of the 5th International Conference on Modeling, Simulation and Applied Optimization. Piscataway: IEEE Press, 2013: 1-6.
    [7] KRICHEN S, DAHMANI N. Solving a load balancing problem with a multi-objective particle swarm optimisation approach: Application to aircraft cargo transportation[J]. International Journal of Operational Research, 2016, 27(1/2): 62. doi: 10.1504/IJOR.2016.078455
    [8] 谷润平, 贾旭颖, 赵向领, 等. 民航货机装载优化准确建模仿真研究[J]. 计算机仿真, 2019, 36(3): 20-26. doi: 10.3969/j.issn.1006-9348.2019.03.005

    GU R P, JIA X Y, ZHAO X L, et al. Research on loading, optimization and accurate modeling and simulation of civil aviation cargo aircraft[J]. Computer Simulation, 2019, 36(3): 20-26(in Chinese). doi: 10.3969/j.issn.1006-9348.2019.03.005
    [9] 赵向领, 杜有权. 基于遗传算法的民用航空器配载问题[J]. 中国科技论文, 2021, 16(8): 849-854. doi: 10.3969/j.issn.2095-2783.2021.08.009

    ZHAO X L, DU Y Q. Civil aircraft stowage based on genetic algorithm[J]. China Science Paper, 2021, 16(8): 849-854(in Chinese). doi: 10.3969/j.issn.2095-2783.2021.08.009
    [10] BROSH I. Optimal cargo allocation on board a plane: A sequential linear programming approach[J]. European Journal of Operational Research, 1981, 8(1): 40-46. doi: 10.1016/0377-2217(81)90027-8
    [11] THOMAS C, CAMPBELL K, HINES G, et al. Airbus packing at federal express[J]. Interfaces, 1998, 28(4): 21-30. doi: 10.1287/inte.28.4.21
    [12] MONGEAU M, BES C. Optimization of aircraft container loading[J]. IEEE Transactions on Aerospace and Electronic Systems, 2003, 39(1): 140-150. doi: 10.1109/TAES.2003.1188899
    [13] KALUZNY B L, SHAW R H A D. Optimal aircraft load balancing[J]. International Transactions in Operational Research, 2009, 16(6): 767-787. doi: 10.1111/j.1475-3995.2009.00723.x
    [14] LIMBOURG S, SCHYNS M, LAPORTE G. Automatic aircraft cargo load planning[J]. Journal of the Operational Research Society, 2012, 63(9): 1271-1283. doi: 10.1057/jors.2011.134
    [15] VERSTICHEL J, VANCROONENBURG W, SOUFFRIAU W, et al. A mixed integer programming approach to the aircraft weight and balance problem[J]. Procedia Social and Behavioral Sciences, 2011, 20: 1051-1059. doi: 10.1016/j.sbspro.2011.08.114
    [16] VANCROONENBURG W, VERSTICHEL J, TAVERNIER K, et al. Automatic air cargo selection and weight balancing: A mixed integer programming approach[J]. Transportation Research Part E:Logistics and Transportation Review, 2014, 65: 70-83. doi: 10.1016/j.tre.2013.12.013
    [17] ZHAO X L, YUAN Y, DONG Y, et al. Optimization approach to the aircraft weight and balance problem with the centre of gravity envelope constraints[J]. IET Intelligent Transport Systems, 2021, 15(10): 1269-1286. doi: 10.1049/itr2.12096
    [18] LARSEN O, MIKKELSEN G. An interactive system for the loading of cargo aircraft[J]. European Journal of Operational Research, 1980, 4(6): 367-373. doi: 10.1016/0377-2217(80)90187-3
    [19] LURKIN V, SCHYNS M. The airline container loading problem with pickup and delivery[J]. European Journal of Operational Research, 2015, 244(3): 955-965. doi: 10.1016/j.ejor.2015.02.027
    [20] BRANDT F. The air cargo load planning problem[D]. Karlsruher: Karlsruher Instituts für Technologie, 2017: 61-70.
    [21] CHEN C S, LEE S M, SHEN Q S. An analytical model for the container loading problem[J]. European Journal of Operational Research, 1995, 80(1): 68-76. doi: 10.1016/0377-2217(94)00002-T
    [22] LIU D S, TAN K C, HUANG S Y, et al. On solving multiobjective bin packing problems using evolutionary particle swarm optimization[J]. European Journal of Operational Research, 2008, 190(2): 357-382. doi: 10.1016/j.ejor.2007.06.032
    [23] YAN S Y, SHIH Y L, SHIAO F Y. Optimal cargo container loading plans under stochastic demands for air express carriers[J]. Transportation Research Part E:Logistics and Transportation Review, 2008, 44(3): 555-575. doi: 10.1016/j.tre.2007.01.006
    [24] LI Y Z, TAO Y, WANG F. A compromised large-scale neighborhood search heuristic for capacitated air cargo loading planning[J]. European Journal of Operational Research, 2009, 199(2): 553-560. doi: 10.1016/j.ejor.2008.11.033
    [25] TANG C H. A scenario decomposition-genetic algorithm method for solving stochastic air cargo container loading problems[J]. Transportation Research Part E:Logistics and Transportation Review, 2011, 47(4): 520-531. doi: 10.1016/j.tre.2010.11.013
    [26] PAQUAY C, SCHYNS M, LIMBOURG S. Three dimensional bin packing problem applied to air cargo[C]//Proceedings of the 4th International Conference on Information Systems, Logistics and Supply Chain. Quebec: Scientific Congresses and Symposiums, 2012: 1-6.
    [27] PAQUAY C, SCHYNS M, LIMBOURG S. A mixed integer programming formulation for the three-dimensional bin packing problem deriving from an air cargo application[J]. International Transactions in Operational Research, 2016, 23(1-2): 187-213. doi: 10.1111/itor.12111
    [28] PAQUAY C, LIMBOURG S, SCHYNS M. A tailored two-phase constructive heuristic for the three-dimensional multiple bin size bin packing problem with transportation constraints[J]. European Journal of Operational Research, 2018, 267(1): 52-64. doi: 10.1016/j.ejor.2017.11.010
    [29] PAQUAY C, LIMBOURG S, SCHYNS M, et al. MIP-based constructive heuristics for the three-dimensional bin packing problem with transportation constraints[J]. International Journal of Production Research, 2018, 56(4): 1581-1592. doi: 10.1080/00207543.2017.1355577
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出版历程
  • 收稿日期:  2022-05-29
  • 录用日期:  2022-08-05
  • 网络出版日期:  2022-08-26
  • 整期出版日期:  2024-04-29

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