A stabilized branch-and-price-and-cut algorithm for the multiple length cutting stock problem

Mathematical models and decomposition algorithms for cutting and packing problems

“…, K. The objective is now to pack all the items into a minimum cost set of bins that respects the availability and the capacity of each bin type. The problem was extensively studied in the literature and various exact and heuristic approaches were proposed, see, e.g., Belov and Scheithauer [29], Alves and Valério de Carvalho [10], and Hemmelmayr et al [152]). …”

“…F AF contains 6 items arcs and 5 loss arcs. To build F RE , (i) we reflect item arcs (0,7) in (0,4) and (4,7) in (4,4); (ii) we remove item arcs (7,10) and (7,11) and loss arcs (7,10) and (10,11); and (iii) we replace loss arc (4,7) with (4,R), thus resulting in 4 item arcs and 3 loss arcs. The reduction is small for this toy instance, but can be impressive for large-size instances (see Section 4.6).…”

“…The second set is a subset of 50 instances proposed in Monacci [220]. Table 4.4 provides the results obtained by the cutting plane algorithm by Belov and Scheithauer [29], the branch-and-price-and-cut by Alves and Valério de Carvalho [10], the variable neighbourhood search by Hemmelmayr et al [152], arc-flow, reflect, and reflect+ with a time limit of 3600 seconds. We reran only [29] on our machine, the other results were copied (and uniformized) from the original papers.…”

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

“…Vanderbeck [9] used integer rounding cuts based on a single row of the original constraints, see also [5]. Such simple cuts were preferred because they do not destroy the structure of the pricing problem.…”

“…Vance [8], Vanderbeck [9], Valério de Carvalho [3,4]. In the latest variant of branch-and-price, Alves and Valério de Carvalho [5] still compute the LP relaxation of the Gilmore-Gomory model in every node. But their branching constraints are based on the x variables.…”