Journal ArticleDOI
An Exact Two-Dimensional Non-Guillotine Cutting Tree Search Procedure
TLDR
A Lagrangean relaxation of a zero-one integer programming formulation of the problem of cutting a number of rectangular pieces from a single large rectangle is developed and used as a bound in a tree search procedure.Abstract:
We consider the two-dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut. We develop a Lagrangean relaxation of a zero-one integer programming formulation of the problem and use it as a bound in a tree search procedure. Subgradient optimization is used to optimize the bound derived from the Lagrangean relaxation. Problem reduction tests derived from both the original problem and the Lagrangean relaxation are given. Incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved.read more
Citations
More filters
Journal ArticleDOI
An improved typology of cutting and packing problems
TL;DR: An improved typology of C&P problems is presented, which is partially based on Dyckhoff’s original ideas, but introduces new categorisation criteria, which define problem categories different from those of Dykhoff.
Journal ArticleDOI
Two-dimensional packing problems: A survey
TL;DR: This work considers problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste by discussing mathematical models, lower bounds, classical approximation algorithms, recent heuristic and metaheuristic methods and exact enumerative approaches.
Journal ArticleDOI
Biased random-key genetic algorithms for combinatorial optimization
TL;DR: This paper presents a tutorial on the implementation and use of biased random-key genetic algorithms for solving combinatorial optimization problems, illustrating the ease in which sequential and parallel heuristics based on biased Random-Key genetic algorithms can be developed.
Journal ArticleDOI
Exact Solution of the Two-Dimensional Finite Bon Packing Problem
Silvano Martello,Daniele Vigo +1 more
TL;DR: This work analyzes a well-known lower bound of the Two-Dimensional Finite Bin Packing Problem, and proposes not lower bounds which are used within a branch-and-bound algorithm for the exact solution of the problem.
Journal ArticleDOI
An analytical model for the container loading problem
Chin-Sheng Chen,S. Lee,Q.S. Shen +2 more
TL;DR: This paper considers the problem of loading containers with cartons of non-uniform size and presents an analytical model to capture the mathematical essence of the problem and extended to formulate some special container loading problems.
References
More filters
Journal ArticleDOI
The Lagrangian Relaxation Method for Solving Integer Programming Problems
TL;DR: This paper is a review of Lagrangian relaxation based on what has been learned in the last decade and has led to dramatically improved algorithms for a number of important problems in the areas of routing, location, scheduling, assignment and set covering.
Journal ArticleDOI
Multistage Cutting Stock Problems of Two and More Dimensions
P. C. Gilmore,Ralph E. Gomory +1 more
TL;DR: In this paper, higher dimensional cutting stock problems are discussed as linear programming problems, and a solution described for the sequencing problem under given simplifying assumptions is given for the auxiliary sequencing problem.
Journal ArticleDOI
Orthogonal Packings in Two Dimensions
TL;DR: Efficient approximation algorithms are devised, their limitations are studied, and worst-case bounds on the performance of the packings they produce are derived.
Journal ArticleDOI
The Theory and Computation of Knapsack Functions
P. C. Gilmore,Ralph E. Gomory +1 more
TL;DR: This paper gives a characterization of knapsack functions and then uses the characterization to develop more efficient methods of computation.
Journal ArticleDOI
Performance Bounds for Level-Oriented Two-Dimensional Packing Algorithms
TL;DR: This work analyzes several “level-oriented” algorithms for packing rectangles into a unit-width, infinite-height bin and gives more refined bounds for special cases in which the widths of the given rectangles are restricted and in which only squares are to be packed.