Section: Application Domains
Cutting and packing problems
A family of problems related to the applications just discussed are cutting and packing problems. In cutting stock problems, one has a supply of large pieces of raw material in stock and a set of demands for small “order” pieces. One must satisfy these demands by cutting the required small pieces out of the large pieces from the stock. The objective is primarily to minimize the waste that is counted as the unused part of used pieces of stock material. A solution is given by a set of feasible cutting patterns, i.e. assortments of order pieces that can be cut out of a given large piece of stock material, such that their accumulated production covers the demands. There are many variants of the cutting stock problem. The main ones concerns the number of significant dimensions of the forms (1D, 2D, 3D, or even 1.5D), specific restrictions on the cutting process, the geometrical arrangements of pieces, and the number of cutting stages. There might be secondary objectives related to the balancing of the workload between different cutting machines, the minimization of the number of different cutting patterns used, or the respect of due dates for instances.
Packing, placement or loading problems can be stated in similar terms. There, one has a set of resources (vehicles, railway cars, machines) that must be packed with items. The objective is to maximize the value of the resulting packing while respecting the capacity of the resources. A solution is given by a set of feasible individual resource packings which together do not pack the same object more than once. Again, many variants exist depending mainly on the number of dimensions in which the capacity of the resources are measured and the specific restrictions on the loading process. Packing problems arise in particular as subproblems in cutting problems since the question of building “good” cutting patterns typically boils down to packing a resource piece with maximum value items. There are many applications of these cutting and packing problems: for instances, the cutting of paper, steel bars, glass, wood, textile, and plastic film; optimization of newspaper layout; scheduling of parallel machines; line balancing; and more generally scheduling problems with limited resources.
In previous work, we developed specialized algorithms for some variants of the knapsack problem that arise as a subproblem in solving cutting stock problems: problems with class bounds [157] or with setup costs [18] . We also set benchmark results for the 1D cutting stock problem using an exact optimization approach based on branch-and-price [160] . We were first to introduce exact algorithm for the 1D problem with setup minimization (a much harder variant) [154] . We also applied a nested decomposition approach to a 2D multi-cutting-stage variant [156] and considered ways of incorporating industrial side-constraints in an exact approach for the 1D problem [140] , [141] . We currently work on 2D orthognal placement problems, combining graph therory and mathematical programmaing approaches.
