In the Two-Bar Charts Packing Problem (2-BCPP), it is required to pack the bar charts (BCs) consisting of two bars into the horizontal unit-height strip of minimal length. The bars may move vertically within the strip, but it is forbidden to change the order and separate the chart's bars. Recently, for this new problem, which is a generalization of the Bin Packing Problem (BPP), Strip Packing Problem (SPP), and 2-Dimensional Vector Packing Problem (2-DVPP), several approximation algorithms with guaranteed estimates were proposed. However, after a preliminary analysis of the solutions constructed by approximation algorithms, we discerned that the guaranteed estimates are inaccurate. This fact inspired us to conduct a numerical experiment in which the approximate solutions are compared to each other and with the optimal ones. To construct the optimal solutions or lower bounds for optimum, we use the Boolean Linear Programming (BLP) formulation of 2-BCPP proposed earlier and apply the CPLEX package. We also use a database of instances for BPP with known optimal solutions to construct the instances for the 2-BCPP with known minimal packing length. The results of the simulation make up the main content of this paper.

中文翻译：

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两柱形图装箱问题的近似算法

在双条形图打包问题 (2-BCPP) 中，需要将包含两个条形的条形图 (BC) 打包成长度最小的水平单位高度条。条形可能会在条带内垂直移动，但禁止更改顺序和分隔图表的条形。最近，对于这个新问题，它是装箱问题 (BPP)、条带包装问题 (SPP) 和二维向量包装问题 (2-DVPP) 的推广，提出了几种具有保证估计的近似算法。然而，在对近似算法构建的解进行初步分析后，我们发现保证估计是不准确的。这一事实激励我们进行数值实验，将近似解相互比较并与最佳解进行比较。为了构建最优解或最优下界，我们使用早先提出的 2-BCPP 的布尔线性规划 (BLP) 公式并应用 CPLEX 包。我们还使用具有已知最优解的 BPP 实例数据库来构建具有已知最小包装长度的 2-BCPP 实例。仿真结果构成了本文的主要内容。