一维下料问题的几点注记

在下料问题中,某一给定规格的物品,在一定的目标和约束条件下,由较大规格的原材料切割而成.本注记主要讨论在2个不同约束和2个不同目标下,一维下料问题所对应的模型之间的关系.2个约束是指等式约束和不等式约束;2个目标包括所用原材料最少和剩下的料头最少.在等式约束下,2个模型是等价的,其对应的连续松弛问题也是等价的.在不等式约束下,2个不同目标所对应的模型是不等价的;但是所用原材料最少为目标的模型的连续问题的最优解也是剩下的料头最少为目标的模型的最优解.

Note on one-dimensional cutting stock problem

In the cutting stock problem (CSP) a given order for smaller pieces has to be cut from larger stock material with some objectives under some constraints. This note discusses the relationships between the models for one-dimensional cutting stock problem (1CSP) under two different constraints and two different objectives.The two constraints are equality and inequality constraints; and the two objectives are to minimize the number and the trim loss of stock material needed to produce the ordered pieces. Under equality constraint, the models with both objectives are equivalent, and their corresponding continuous relaxation problems are also equivalent.Under inequality constraint, the models with these two objectives are not equivalent, but the optimal solutions of the corresponding continuous relaxation problem for the number-minimizing model are also optimal for the trim-loss-minimizing model.