A Nested Decomposition Approach to a Three-Stage, Two-Dimensional Cutting-Stock Problem
提出一种嵌套分解方法，递归使用列生成技术求解三阶段二维下料问题，目标是最小化废料并处理库存老化、紧急订单等实际约束，算法已在工业中用于生产计划。
We consider the cutting of rectangular order pieces into stock pieces of specified width and length. The cutting process involves three stages of orthogonal guillotine cutting: Stock pieces are cut into sections that are cut into slits that are cut into order pieces. Restrictions imposed on the cutting process make the combinatorial structure of the problem more complex, but limit the scope of solution space. The objective of the problem is mainly to minimize waste, but our model also accounts for other issues such as aging stock pieces, urgent or optional orders, and fixed setup costs. Our solution approach involves a nested decomposition of the problem and the recursive use of the column-generation technique: We use a column-generation formulation of the problem (Gilmore and Gomory 1965) and the cutting-pattern—generation subproblem is itself solved using a column-generation algorithm. LP-based lower bounds on the minimum cost are computed and, by rounding the LP solution, a feasible solution and associated upper bound is obtained. This approach could in principle be used in a branch-and-bound search to solve the problem to optimality. We report computational results for industrial instances. The algorithm is being used in industry as a production-planning tool.