The one dimensional cutting stock problem (1D-CSP) is one of the representative combinatorial optimization problems, which arises in many industries. As the setup costs of cutting patterns become more dominant in recent cutting industry, we consider a variant of 1D-CSP, in which the total number of applications of cutting patterns is minimized under the constraint that the number of different cutting patterns is specified in advance. We propose a local search algorithm that uses the neighborhood obtained by perturbating one cutting pattern in the current set of patterns, where the perturbations are done by utilizing the dual solution of the auxiliary linear programming problem (LP). In this process, in order to solve a large number of LPs, we start the criss-cross variation of the simplex algorithm from the optimal simplex tableau of the previous solution, instead of starting it from scratch. According to our computational experiment, it is observed that the proposed algorithm obtains a wide variety of good solutions which are comparable to the existing heuristic approaches.

We consider a data set in which each example is an n-dimensional Boolean vector labeled as true or false. A pattern is a co-occurrence of a particular value combination of a given subset of the variables. If a pattern appears frequently in the true examples and infrequently in the false examples, we consider it a good pattern. In this paper, we discuss the problem of determining the data size needed for removing "deceptive" good patterns; in a data set of a small size, many good patterns may appear superficially, simply by chance, independently of the underlying structure. Our hypothesis is that, in order to remove such deceptive good patterns, the data set should contain a greater number of examples than that at which a random data set contains few good patterns. We justify this hypothesis by computational studies. We also derive a theoretical upper bound on the needed data size in view of our hypothesis.