This paper presents theoretical and practical results for the bin packing problem with scenarios, a generalization of the classical bin packing problem which considers the presence of uncertain scenarios, of which only one is realized. For this problem, we propose approximation algorithms whose ratios are bounded by the square root of the number of scenarios times the approximation ratio for an algorithm for the vector bin packing problem. We also show how an asymptotic polynomial-time approximation scheme is derived when the number of scenarios is constant, that is, not a part of the input. As a practical study of the problem, we present a branch-and-price algorithm to solve an exponential set-cover model and a variable neighborhood search heuristic. Experiments show the competence of the branch-and-price in obtaining optimal solutions for about 59% of the instances considered, while the combined heuristic and branch-and-price optimally solved 62% of the instances considered.

本文介绍了具有情景的 bin 包装问题的理论和实践结果，这是经典 bin 包装问题的推广，该问题考虑了不确定场景的存在，其中只实现了一种。对于这个问题，我们提出了近似算法，其比率以场景数的平方根乘以向量分箱打包问题算法的近似比率为界。我们还展示了当场景数量恒定（即不是输入的一部分）时如何推导出渐近多项式时间近似方案。作为对该问题的实际研究，我们提出了一种 branch-and-price 算法来解决指数集合覆盖模型和可变邻域搜索启发式算法。实验表明，branch-and-price 能够为大约 59% 的所考虑实例获得最优解决方案，而启发式和 branch-and-price 的组合以最佳方式解决了 62% 的所考虑实例。