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The problem addressed in this paper is that of orthogonally packing a given set of rectangular-shaped items into the minimum number of three-dimensional rectangular bins. The problem is strongly NP-hard and extremely di cult to solve in practice. Lower bounds are discussed, and it is proved that the asymptotic worst-case performance ratio of the continuous lower bound is 1 8. An exact algorithm for lling a single bin is developed, leading to the de nition of an exact branch-and-bound algorithm for the three-dimensional bin packing problem, which also incorporates original approximation algorithms. Extensive computational results, involving instances with up to 90 items, are presented: it is shown that many instances can be solved to optimality within a reasonable time limit.

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Two-dimensional bin packing problems consist of allocating, without overlapping, a given set of small rectangles (items) to a minimum number of large identical rectangles (bins), with the edges of the items parallel to those of the bins. According to the specific application, the items may either have a fixed orientation or they can be rotated by 90°. In addition, it may or not be imposed that the items are obtained through a sequence of edge-to-edge cuts parallel to the edges of the bin. In this article, we consider the class of problems arising from all combinations of the above requirements. We introduce a new heuristic algorithm for each problem in the class, and a unified tabu search approach that is adapted to a specific problem by simply changing the heuristic used to explore the neighborhood. The average performance of the single heuristics and of the tabu search are evaluated through extensive computational experiments.

Given a set of rectangular pieces to be cut from an unlimited number of standardized stock pieces (bins), the Two-Dimensional Finite Bin Packing Problem is to determine the minimum number of stock pieces that provide all the pieces. The problem is NP-hard in the strong sense and finds many practical applications in the cutting and packing area. We analyze a well-known lower bound and determine its worst-case performance. We propose new lower bounds which are used within a branch-and-bound algorithm for the exact solution of the problem. Extensive computational testing on problem instances from the literature involving up to 120 pieces shows the effectiveness of the proposed approach.Cutting and Packing, Branch-and-Bound, Worst-Case Analysis

Two new algorithms recently proved to outperform all previous methods for the exact solution of the 0-1 Knapsack Problem. This paper presents a combination of such approaches, where, in addition, valid inequalities are generated and surrogate relaxed, and a new initial core problem is adopted. The algorithm is able to solve all classical test instances, with up to 10,000 variables, in less than 0.2 seconds on a HP9000-735/99 computer. The C language implementation of the algorithm is available on the internet.Knapsack problem, dynamic programming, branch-and-bound, surrogate relaxation