Multi-variant Optimization Algorithm for Three Dimensional Container Loading Problem
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摘要: 用多元优化算法（Multi-variant optimization algorithm，MOA）实现三维装箱问题的求解.算法通过随机放置和局部调整从而逐步逼近最优解.随机放置是将随机选择的几个箱子装入容器中；局部调整是根据目标函数值对随机放置容器的箱子序列作局部调整优化；通过递推的随机放置和局部调整优化，目标函数值逐步逼近最优值，从而获得一个较为理想的三维装箱方案.算法通过对BR1~BR10共1000组三维装箱问题测试实例的测试仿真，得到理想的装箱效果，说明用多元优化算法实现三维装箱问题的有效性和可行性.Abstract: This paper investigates that three-dimensional container loading problem is solved by the Multi-variant optimization algorithm. The Multi-variant optimization algorithm applied random placement and partial adjustment to gradually approximate the optimal solution. The random placement denotes that several boxes of randomly selected are put into the container; the partial adjustment indicates that the sequence of the boxes in the container of random placement are topically adjusted and optimized with objective function value. Then, the objective function value will gradually approximate the optimal value by recursively random replacement, partial adjustment and optimization, and we acquire a desirable three-dimensional container loading program. In order to verify the effectiveness and practicability of three-dimensional container problem with the multi-variant optimization algorithm, 1000 groups of three-dimensional container loading problems that vary from BR1-BR10 are tested in this paper and acquire desirable results.1) 本文责任编委 王红卫
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图 7 MOA实现装箱问题过程中填充率变化图
Fig. 7 The filling rate variation diagram of packing problem with MOA
表 1 BR1-1待装箱子的三维值及数量
Table 1 The specification and quantity of BR1-1
长(cm) 宽(cm) 高(cm) 数量 108 76 30 40 110 43 25 33 92 81 55 39 
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表 2 某次实验得到的填充率的变化关系表
Table 2 The change table of filling rate obtained from an experiment
批次 1/(调整优化后) 2/(调整优化后) 3/(调整优化后) 4/(调整优化后) 5/(调整优化后) 容器填充率 0.2749/(0.3591) 0.5495/(0.5968) 0.7324/(0.7936) 0.8926/(0.9065) 0.9625/(0.9625)
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表 3 各种算法的装箱效果比较(1 ~ 500组)
Table 3 Comparison of packing effects of various (groups 1 ~ 500) algorithms
测试实例 约束 填充率(%) BR1 BR2 BR3 BR4 BR5 箱子的种类数 3 5 8 10 12 H_BR[3] C1 and C2 83.79 84.44 83.94 83.71 83.80 GA_GB[6] C1 and C2 85.80 87.26 88.10 88.04 87.86 TS_BG[7] C1 and C2 87.81 89.40 90.48 90.63 90.73 GRASP[12] C1 93.52 93.77 93.58 93.05 92.34 Maximal-space[15] C1 93.85 94.22 94.25 94.09 93.87 HSA[17] C1 and C2 93.81 93.94 93.86 93.57 93.22 CLTRS[18] C1 95.05 95.43 95.47 95.18 95.00 C1 and C2 94.51 94.73 94.73 94.41 94.13 MLHS[19] C1 94.92 95.48 95.69 95.53 95.44 C1 and C2 94.49 94.89 95.20 94.94 94.78 VNS[24] C1 94.93 95.19 94.99 94.71 94.33 FDA[25] C1 92.92 93.93 93.71 93.68 93.73 MOA C1 and C2 95.62 94.68 94.41 94.23 94.03
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表 4 各种算法的装箱效果比较(501 ~ 1 000)
Table 4 Comparison of packing effects of various (groups 501 ~ 1 000) algorithms
测试实例 约束 填充率(%) BR6 BR7 BR8 BR9 BR10 平均 箱子的种类数 15 20 30 40 50 H_BR[3] C1 and C2 82.44 82.01 80.10 78.03 76.53 81.88 GA_GB[6] C1 and C2 87.85 87.68 87.52 86.46 85.53 87.21 TS_BG[7] C1 and C2 92.72 90.65 87.11 85.76 84.73 89.00 GRASP[12] C1 91.72 90.55 86.13 85.08 84.21 90.40 Maximal-space[15] C1 93.52 92.94 91.02 90.46 89.87 92.81 HSA[17] C1 and C2 92.72 91.99 90.56 89.70 89.06 92.24 CLTRS[18] C1 94.79 94.24 93.70 93.44 93.09 94.54 C1 and C2 93.85 93.20 92.26 91.48 90.86 93.42 MLHS[19] C1 95.38 94.95 94.54 94.14 93.95 95.00 C1 and C2 94.55 93.95 93.12 92.48 91.83 94.02 VNS[24] C1 94.04 93.53 92.78 92.19 91.92 93.86 FDA[25] C1 93.63 93.14 92.92 92.49 92.24 93.23 MOA C1 and C2 94.56 93.27 93.09 91.52 91.00 93.64
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表 5 元优化算法实现三维装箱问题时的一些测试细节
Table 5 Some test details of MOA for 3D packing problem
测试实例 约束 运行时间(s) 填充率(%) Minimum Maximum Average Minimum Maximum Average BR1 3 1.83 114.66 28.84 91.04 98.31 95.62 BR2 5 2.41 57.89 26.78 89.18 97.23 94.68 BR3 8 3.42 191.23 86.35 90.60 96.97 94.41 BR4 10 1.26 274.91 105.67 88.82 96.04 94.23 BR5 12 7.50 219.01 110.63 89.17 97.45 94.03 BR6 15 5.83 495.02 265.74 87.36 96.56 94.56 BR7 20 14.61 811.50 384.38 5.82 95.44 93.27 BR8 30 30.82 1 312.80 560.21 84.61 95.17 93.09 BR9 40 33.40 1 798.75 866.08 84.69 94.07 91.52 BR10 50 54.52 2 401.94 1 380.62 82.27 93.84 91.00 Mean 19.30 15.56 767.77 381.53 87.36 96.11 93.64
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