Integer and combinatorial optimization
Integer and combinatorial optimization
Knapsack problems: algorithms and computer implementations
Knapsack problems: algorithms and computer implementations
Multilinear programming: duality theories
Journal of Optimization Theory and Applications
Families of non-IRUP instances of the one-dimensional cutting stock problem
Discrete Applied Mathematics
New Classes of Lower Bounds for Bin Packing Problems
Proceedings of the 6th International IPCO Conference on Integer Programming and Combinatorial Optimization
Mathematical Programming: Series A and B
A Combinatorial Characterization of Higher-Dimensional Orthogonal Packing
Mathematics of Operations Research
Computers and Operations Research
INFORMS Journal on Computing
A new constraint programming approach for the orthogonal packing problem
Computers and Operations Research
Bin packing with items uniformly distributed over intervals [a,b]
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Large gaps in one-dimensional cutting stock problems
Discrete Applied Mathematics
Bidimensional packing by bilinear programming
Mathematical Programming: Series A and B
An Exact Algorithm for Higher-Dimensional Orthogonal Packing
Operations Research
New filtering for the cumulative constraint in the context of non-overlapping rectangles
CPAIOR'08 Proceedings of the 5th international conference on Integration of AI and OR techniques in constraint programming for combinatorial optimization problems
An Exact Algorithm for the Two-Dimensional Strip-Packing Problem
Operations Research
New Stabilization Procedures for the Cutting Stock Problem
INFORMS Journal on Computing
Theoretical investigations on maximal dual feasible functions
Operations Research Letters
On the two-dimensional Knapsack Problem
Operations Research Letters
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Packing problems (sometimes also called cutting problems) are combinatorial optimization problems concerned with placement of objects (items) in one or several containers. Some packing problems are special cases of several other problems such as resource-constrained scheduling, capacitated vehicle routing, etc. In this paper we consider a bounding technique for one- and higher-dimensional orthogonal packing problems, called conservative scales (CS) (in the scheduling terminology, redundant resources). CS are related to the possible structure of resource consumption: filling of a bin, distribution of the resource to the jobs, etc. In terms of packing, CS are modified item sizes such that the set of feasible packings is not reduced. In fact, every CS represents a valid inequality for a certain binary knapsack polyhedron. CS correspond to dual variables of the set-partitioning model of a special 1D cutting-stock problem. Some CS can be constructed by (data-dependent) dual-feasible functions ((D)DFFs). We discuss the relation of CS to DFFs: CS assume that at most one copy of each object can appear in a combination, whereas DFFs allow several copies. The literature has investigated the so-called extremal maximal DFFs (EMDFFs) which should provide very strong CS. Analogously, we introduce the notions of maximal CS (MCS) and extremal maximal CS (EMCS) and show that EMDFFs do not necessarily produce (E)MCS. We propose fast greedy methods to "maximize" a given CS. Using the fact that EMCS define facets of the binary knapsack polyhedron, we use lifted cover inequalities as EMCS. For higher-dimensional orthogonal packing, we propose a Sequential LP (SLP) method over the set of CS and investigate its convergence. Numerical results are presented.