One-and two-dimensional packing and cutting problems occur in many commercial contexts, and it is often important to be able to get good-quality solutions quickly. Fairly simple deterministic heuristics are often used for this purpose, but such heuristics typically find excellent solutions for some problems and only mediocre ones for others. Trying several different heuristics on a problem adds...

In this paper we consider a simplified version of the stock cutting (two-dimensional bin packing) problem. We compare three meta-heuristic algorithms (genetic algorithm (GA), tabu search (TS) and simulated annealing (SA)) when applied to this problem. The results show that tabu search and simulated annealing produce good quality results. This is not the case with the genetic algorithm. The prob...

The multidimensional knapsack problem is defined as an optimization problem that is NP-hard combinatorial. The multidimensional knapsack problems have large applications, which include many applicable problems from different area, like cargo loading, cutting stock, bin-packing, financial and other management, etc. This paper reviews some researches published in the literature. The concentrate i...

In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, these classes of cutting planes must necessarily be applied for many r...

Packing Three-Dimensional Irregular Objects Because of its many applications in practice, the cutting and packing literature is extensive well established. It mostly concerned with problems one two dimensions or where some regularity pieces assumed (e.g., boxes). However, rise realm three-dimensional printing additive manufacturing has created a demand for efficient irregular objects. In “Voxel...

In this paper, we address the combined assortment and trim loss minimization problem. In real settings, when different stock lengths are available, a common problem is to select a subset of stock lengths from which to cut the ordered items. In the cutting and packing literature, this problem is known as the assortment problem. Solving the assortment and trim loss minimization problem in a singl...

Rectangle-packing problem involves many industrial applications. For example, in shipping industry, various size boxes have to be loaded as many as possible in a larger container. In wood or glass industries, rectangular components have to be cut from large sheets of material. In very large scale integration (VLSI) floor planning, various chips have to be laid on the chip board, and so on. The ...

We consider two (0,1)-linear programming formulations of the graph (vertex-) coloring problem, in which variables are associated to stable sets of the input graph. The first one is a set covering formulation, where the set of vertices has to be covered by a minimum number of stable sets. The second is a set packing formulation, in which constraints express that two stable sets cannot have a com...