We study a robust extensible bin packing problem with budgeted uncertainty, under a budgeted uncertainty model where item sizes are defined to lie in the intersection of a box with a one-norm ball. We propose a scenario generation algorithm for this problem, which alternates between solving a master robust bin-packing problem with a finite uncertainty set and solving a separation problem. We first show that the separation is strongly NP-hard given solutions to the continuous relaxation of the master problem. Then, focusing on the separation problem for the integer master problem, we show that this problem becomes a special case of the continuous convex knapsack problem, which is known to be weakly NP-hard. Next, we prove that our special case when each of the functions is piecewise linear, having only two pieces, remains NP-hard. We develop a pseudo-polynomial dynamic program (DP) and a fully polynomial-time approximation scheme (FPTAS) for our special case whose running times match those of a binary knapsack FPTAS. Finally, our computational study shows that the DP can be significantly more efficient in practice compared with solving the problem with specially ordered set (SOS) constraints using advanced mixed-integer (MIP) solvers. Our experiments also demonstrate the application of our separation problem method to solving the robust extensible bin packing problem, including the evaluation of deferring the exact solution of the master problem, separating based on approximate master solutions in intermediate iterations. Finally, a case-study, based on real elective surgery data, demonstrates the potential advantage of our model compared with the actual schedule and optimal nominal schedules.

翻译：本文研究一种基于预算不确定性的鲁棒可扩展装箱问题，其中物品尺寸被定义为箱型约束与一范数球交集中的取值。我们提出了一种适用于该问题的场景生成算法，该算法交替求解具有有限不确定集的主鲁棒装箱问题与分离问题。首先，我们证明在给定主问题连续松弛解的情况下，分离问题是强NP难的。随后，聚焦于整数主问题的分离问题，我们证明该问题可转化为连续凸背包问题的特例，而后者已知为弱NP难问题。进一步，我们证明当每个函数均为仅含两段的分段线性函数时，该特例仍保持NP难性。针对此特例，我们开发了一种伪多项式动态规划算法及完全多项式时间近似方案，其运行时间与二元背包问题的FPTAS相匹配。最后，计算研究表明，与使用先进混合整数规划求解器处理带特殊有序集约束的模型相比，动态规划算法在实际应用中可显著提升效率。实验还展示了我们的分离问题方法在求解鲁棒可扩展装箱问题中的应用，包括评估在中间迭代中基于近似主问题解进行分离、延迟主问题精确求解的策略。基于真实择期手术数据的案例研究表明，与现行排程及最优名义排程相比，我们的模型具有潜在优势。