Constructing an instance of the cutting stock problem of minimum size which does not possess the integer round-up property

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 $L \le 15$ 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 $L=18$. Tab. 2, bibliogr. 14.