The proper relaxation and the proper gap of the skiving stock problem

Weconsider the 1D skiving stock problem (SSP)which is strongly related to the dual bin packing problem: find the maximum number of products 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 NP-hard optimization problem, we focus on the proper relaxation and introduce a modeling approach based on graph theory. Additionally, we investigate the quality of the proper gap, i.e., the difference between the optimal objective values of the proper relaxation and the SSP itself. As an introductorily motivation, we prove that the SSP does not possess the integer round down property (IRDP) with respect to the proper relaxation. The main contribution of this paper is given by a construction principle for an infinite number of non-equivalent non-proper-IRDP instances and an enumerative approach that leads to the currently largest known (proper) gap.