6. Cutting and Packing Problems with Placement Constraints

In real-life problems of cutting and packing very often placement constraints are present. For instance, defective regions of the raw material (wooden boards, steel plates, etc.) shall not become part of the desired products. More generally, due to different quality demands, some products may contain parts of lower quality which are not allowed for other goods. Within this work we consider one- and two-dimensional rectangular cutting and packing problems where items of given types have to be cut from (or packed on) raw material such that an objective function attains its maximum. In the one-dimensional (1D) case, we assume for each item type that allocation intervals (regions of the raw material) are given so that any item of the same type must be completely contained in one of the corresponding allocation intervals. In addition, we deal with problems where the lengths of the 1D items of a given type may vary within known tolerances. In the two-dimensional (2D) case, where rectangular items of different types have to be cut from a large rectangle, we investigate guillotine cutting under the condition that defective rectangular regions are not allowed to be part of the manufactured products (even not partially). For these scenarios we present solution strategies which rely on the branch and bound principle or on dynamic programming. Based on properties of the corresponding objective functions we discuss possibilities to reduce the computational complexity. This includes the definition of appropriate sets of potential allocation (cut) points which have to be inspected to obtain an optimal solution. By dominance considerations the set of allocation points is kept small. In particular, the computational complexity becomes independent of the unit of measure of the input data. Possible generalizations will be discussed as well.