Gomory Cuts from a Position-Indexed Formulation of 1D Stock Cutting

Abstract

Abstract. Most integer programming problems can be formulated in several ways. Some formulations are better suited for solution by exact methods, because they have either (i) a strong LP relaxation, (ii) few symmetries in the solution space, or both. However, solving one formulation, we can often branch and/or add cutting planes which are implicitly based on variables of other formulations, working in fact on intersection of several polytopes. Traditional examples of this approach can be found in, e.g., (capacitated) routing and network planning where decomposed models operate with paths or trees, and thus need to be solved by column generation, but original models operate on separate edges. We consider such a 'capacity-extended formulation', the so-called arc-flow model, of the 1D cutting stock problem. Its variables are known to induce effective branching constraints leading to small and stable branch&bound trees. In this work we explore Chvátal-Gomory cuts on its variables. The results are positive only for small instances. Moreover, we compare the results to the cuts constructed on the variables of the direct model. The latter are more involved but also more effective.

Mathematical models and decomposition algorithms for cutting and packing problems

“…The resulting algorithm directly branches on the variables associated with the patterns, selecting the variable whose fractional value is closer to 0.5. Later on, Belov et al [31] investigated the performance of combining Chvátal-Gomory cuts and arc-flow formulations, which however did not prove to be very effective.…”

A stabilized structured Dantzig–Wolfe decomposition method

“…For instance, the well-known Gilmore-Gomory formulation of the cutting stock problem [8] is usually solved by DW approaches; there, X is the set of all valid cutting patterns, that is, all feasible solutions to an integer knapsack problem. Owing to the well-known reformulation of integer knapsack problems in terms of longest path problems on a directed acyclic network, one can devise the arc-flow model of the cutting stock problem [5], that provides the same lower bound at the cost of a pseudo-polynomial number of variables (one for each arc in the graph representing the knapsack) and constraints (one for each node in the same graph). The arc-flow model provides the alternative reformulation of Assumption 1, and each feasible cutting pattern is a path in the directed graph underlying the arc-flow model; thus, one can easily devise a restricted formulation X B corresponding to a (small) sub-graph of the (large) full directed graph.…”