Constrained two‐dimensional cutting stock problems a best‐first branch‐and‐bound algorithm

Abstract

In this paper, we develop a new version of the algorithm proposed in Hifi (Computers and Operations Research 24/8 (1997) 727–736) for solving exactly some variants of (un)weighted constrained two‐dimensional cutting stock problems. Performance of branch‐and‐bound procedure depends highly on particular implementation of that algorithm. Programs of this kind are often accelerated drastically by employing sophisticated techniques. In the new version of the algorithm, we start by enhancing the initial lower bound to limit initially the space search. This initial lower bound has already been used in Fayard et al. 1998 (Journal of the Operational Research Society, 49, 1270–1277), as a heuristic for solving the constrained and unconstrained cutting stock problems. Also, we try to improve the upper bound at each internal node of the developed tree, by applying some simple and effcient combinations. Finally, we introduce some new symmetric‐strategies used for neglecting some unnecessary duplicate patterns. The performance of our algorithm is evaluated on some problem instances of the literature and other hard‐randomly generated problem instances.

Constrained two‐dimensional guillotine cutting problem: upper‐bound review and categorization

“…Cung et al. () added further improvements in lower and upper bounds, besides a parallelization of the B&B. Three antiredundancy rules were also added.…”

“…Starting from the work by Cung et al. (), several authors used the strategy referred to as D1. This strategy exploits the fact that a new build obtained by merging B and B′ has to be dropped if a piece ψ of type t i , such that d i > b i + b′ i , can be added in the waste top‐right zone (gray area in Fig.…”

“…In this way, the authors replaced other more complex strategies proposed by Cung et al. () affected by further drawbacks on the accuracy. It is important to note that this check can also be interpreted as the generalization to the constrained case of the check performed by Young‐Gun et al.…”

“…() proposed the best exact bottom‐up algorithm, but they kept the inaccurate strategy of Cung et al. (). This algorithm is a modification of the algorithm proposed by Viswanathan and Bagchi (), which improved in two aspects.…”

A heuristic approach based on dynamic programming and and/or-graph search for the constrained two-dimensional guillotine cutting problem

“…For example, Christofides and Whitlock (1977) presented a tree search algorithm with dynamic programming and a transportation routine, Viswanathan and Bagchi (1993) proposed a best-first search method based on Wang (1983)'s bottom-up approach, Hifi (1997a) presented an improvement of Viswanathan and Bagchi (1993)'s algorithm, and Cung et al (2000) presented a new version of Hifi (1997a)'s procedure. In particular, Christofides and Hadjiconstantinou (1995) developed a tree search algorithm using bounds derived from a state space relaxation of a dynamic programming formulation and a state space ascent procedure, which is an improvement of Christofides and Whitlock (1977)'s algorithm.…”