This work presents an empirical analysis of exact algorithms for the unbounded knapsack problem, which includes seven algorithms from the literature, two commercial solvers, and more than ten thousand instances. The terminating stepoff, a dynamic programming algorithm from 1966, presented the lowest mean time to solve the most recent benchmark from the literature. The threshold and collective dominances are properties of the unbounded knapsack problem first discussed in 1998, and exploited by the current state-of-the-art algorithms. The terminating step-off algorithm did not exploit such dominances, but has an alternative mechanism for dealing with dominances which does not explicitly exploits collective and threshold dominances. Also, the pricing subproblems found when solving hard cutting stock problems with column generation can cause branch-and-bound algorithms to display worst-case times. The authors present a new class of instances which favors the branch-and-bound approach over the dynamic programming approach but do not have high amounts of simple, multiple and collective dominated items. This behaviour illustrates how the definition of hard instances changes among algorithm approachs. The codes used for solving the unbounded knapsack problem and for instance generation are all available online.

No abstract

We advance the state of the art in Mixed-Integer Linear Programming formulations for Guillotine 2D Cutting Problems by (i) adapting a previously-known reduction to our preprocessing phase (plate-size normalization) and by (ii) enhancing a previous formulation (PP-G2KP from Furini et alli) by cutting down its size and symmetries. Our focus is the Guillotine 2D Knapsack Problem with orthogonal and unrestricted cuts, constrained demand, unlimited stages, and no rotation – however, the formulation may be adapted to many related problems. The code is available. Concerning the set of 59 instances used to benchmark the original formulation, the enhanced formulation takes about 4 hours to solve all instances while the original formulation takes 12 hours to solve 53 of them (the other six runs hit a three-hour time limit each). We integrate, to both formulations, a pricing framework proposed for the original formulation; the enhanced formulation keeps a significant advantage in this situation. Finally, in a recently proposed set of 80 harder instances, the enhanced formulation (with and without the pricing framework) found: 22 optimal solutions (5 already known, 17 new); better lower bounds for 25 instances; better upper bounds for 58 instances.

No abstract

About 15 years ago, a paper proposed the first integer linear programming formulation for the constrained two‐dimensional guillotine cutting problem (with unlimited cutting stages). Since then, eight other formulations followed, seven of them in the last four years. This spike of interest gave no opportunity for a comprehensive comparison between the formulations. We review each formulation and compare their empirical results over instance datasets of the literature. We adapt most formulations to allow for piece rotation. The possibility of adaptation was already predicted but not realized by the prior work. The results show the dominance of pseudo‐polynomial formulations until the point instances become intractable by them, while more compact formulations keep achieving good primal solutions. Our study also reveals a mistake in the generation of the T instances, which should have the same optima with or without guillotine cuts. We also propose hybridising a recent formulation with a prior formulation for a restricted version of the problem. The hybridisations show a reduction of about 20% of the branch‐and‐bound time thanks to the symmetries broken by the hybridisation.