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 an absolute approximation algorithm whose ratio is 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. As a practical study of the problem, we present a branch-and-price algorithm to solve an exponential model and a variable neighborhood search heuristic. To speed up the convergence of the exact algorithm, we also consider lower bounds based on dual feasible functions. Results of these algorithms 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.

翻译：本文针对带有场景的装箱问题（经典装箱问题考虑不确定场景存在且仅实现其中一种场景的推广）提出了理论与实际结果。针对该问题，我们提出了一种绝对近似算法，其比率受场景数量的平方根乘以向量装箱问题算法的近似比约束。我们还证明了在场景数量为常数时，如何推导出渐近多项式时间近似方案。作为该问题的实践研究，我们提出了一种分支定价算法以求解指数模型，以及一种可变邻域搜索启发式算法。为加速精确算法的收敛，我们还考虑了基于对偶可行函数的下界。这些算法的结果表明，分支定价在约59%的测试实例中获得最优解，而启发式与分支定价的组合方法在62%的测试实例中获得最优解。