Two-dimensional skiving and cutting stock problem with setup cost based on column-and-row generation

In this paper, we address a new variant of the cutting stock problem, in which skiving is allowed and setup costs are considered. Specifically, the output sheets are generally longer but narrower than the input coils. To satisfy the demand of each sheet, combining two or more coils is allowed. Moreover, because changing from one pattern to another involves considerable time and cost, minimizing the number of different patterns or setups is vital. Thus, the objective of our study is to minimize the material cost and the number of setups. We propose an integer programming (IP) formulation for the problem that contains an exponential number of binary variables and column-dependent constraints. The linear programming (LP) relaxation is solved using a column-and-row generation framework that involves a knapsack subproblem and a nonlinear IP subproblem. For the latter, we propose a decomposition-based exact solution method with pseudo-polynomial time. To obtain IP solutions, we develop a diving heuristic based on matching level. The computational experiments show that these algorithms are efficient and of high quality.