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.

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