Integer rounding and modified integer rounding for the skiving stock problem

We consider the one-dimensional skiving stock problem which is also known as the dual bin packing problem: find the maximum number of items with minimum length L that can be constructed by connecting a given supply of m ∈ N smaller item lengths l1, . . . , lm with availabilities b1, . . . , bm. For this optimization problem, we investigate the quality of the continuous relaxation by considering the gap, i.e., the difference between the optimal objective values of the continuous relaxation and the skiving stock problem itself. In a first step, we derive an upper bound for the gap by generalizing a result of E. J. Zak. As a main contribution, we prove the modified integer round-down property of the divisible case. In this context, we also present a construction principle for non-IRDP instances of the divisible case and investigate the maximum gap that can be obtained based on these instances.