We consider the well-known one-dimensional cutting stock problem in order to find some integer instances with the minimal length L of a stock material for which the round-up property is not satisfied. The difference between the exact solution of an instance of a cutting stock problem and the solution of its linear relaxation is called the integrality gap. Some instance of a cutting problem has the integer round-up property (IRUP) if its integrality gap is less than 1. We present a new method for exhaustive search over the instances with maximal integrality gap when the values of L, the lengths of demanded pieces, and the optimal integer solution are fixed. This method allows us to prove by computing that all instances with L ≤ 15 have the round-up property. Also some instances are given with L = 16 not-possessing this property, which gives an improvement of the best known result L = 18.

中文翻译：

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构造不具有整数取整属性的最小尺寸的切削刀具问题的实例

我们考虑众所周知的一维切削原料问题，以便找到一些具有最小材料长度L的整数实例，而该整数实例的舍入性能不满足要求。一个切削问题的精确解与线性松弛的解之间的差异被称为完整性差距。如果某个切削问题的实例的整数间隙小于1，则该实例具有整数舍入属性（IRUP）。当L的值（即所需长度）为最大值时，我们提出了一种新的穷举搜索方法，用于对实例进行穷举搜索。件，并且最佳整数解是固定的。这种方法使我们能够通过计算来证明大号≤15有轮动性。还给出了某些实例，其中L = 16不具有此属性，从而改善了最著名的结果L = 18。