An Analysis of Solutions to the 2D Bin Packing Problem and Additional Complexities (original) (raw)

The Two-Dimensional Finite Bin Packing Problem. Part II: New lower and upper bounds

Quarterly Journal of the Belgian, French and Italian Operations Research Societies, 2003

The Two-Dimensional Finite Bin Packing Problem (2BP) consists of determining the minimum number of large identical rectangles, bins, that are required for allocating without overlapping a given set of rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. In this paper we describe new lower bounds for the 2BP where the items have a fixed orientation and we show that the new lower bounds dominate two lower bounds proposed in the literature. These lower bounds are extended in Part II (see ) for a more general version of the 2BP where some items can be rotated by 90 • . Moreover, in Part II a new heuristic algorithm for solving both versions of the 2BP is presented and computational results on test problems from the literature are given in order to evaluate the effectiveness of the proposed lower bounds.

The two-dimensional finite bin packing problem. Part I: New lower bounds for the oriented case

4OR, 2003

The Two-Dimensional Finite Bin Packing Problem (2BP) consists of determining the minimum number of large identical rectangles, bins, that are required for allocating without overlapping a given set of rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. In this paper we describe new lower bounds for the 2BP where the items have a fixed orientation and we show that the new lower bounds dominate two lower bounds proposed in the literature. These lower bounds are extended in Part II (see Boschetti and Mingozzi (2002)) for a more general version of the 2BP where some items can be rotated by 90 •. Moreover, in Part II a new heuristic algorithm for solving both versions of the 2BP is presented and computational results on test problems from the literature are given in order to evaluate the effectiveness of the proposed lower bounds.

Multidimensional Bin Packing and Other Related Problems : A Survey ∗

2016

The bin packing problem is a well-studied problem in combinatorial optimization. In the classical bin packing problem, we are given a list of real numbers in (0, 1] and the goal is to place them in a minimum number of bins so that no bin holds numbers summing to more than 1. The problem is extremely important in practice and finds numerous applications in scheduling, routing and resource allocation problems. Theoretically the problem has rich connections with discrepancy theory, iterative methods, entropy rounding and has led to the development of several algorithmic techniques. In this survey we consider several classical generalizations of bin packing problem such as geometric bin packing, vector bin packing and various other related problems. In two-dimensional geometric bin packing, we are given a collection of rectangular items to be packed into a minimum number of unit size square bins. This variant has a lot of applications in cutting stock, vehicle loading, pallet packing, m...

On a dual version of the one-dimensional bin packing problem

Journal of Algorithms, 1984

The NP-hard problem of packing items from a given set into bins so as to maximize the number of bins used, subject to the constraint that each bin be filled to at least a given threshold, is considered. Approximation algorithms are presented that provide guarantees of i, i, and i the optimal number, at running time costs of O(n), 0( n log n), and O(n log2n), respectively, and the average case behavior of these algorithms is explored via empirical tests on randomly generated sets of items, T'

A new exact method for the two-dimensional bin-packing problem with fixed orientation

Operations Research Letters, 2007

We propose a new exact method for the well-known two-dimensional bin-packing problem. It is based on an iterative decomposition of the set of items into two disjoint subsets. We tested the efficiency of our method against benchmarks of the literature. Computational experiments confirm the efficiency of our method.

Algorithms for the two dimensional bin packing problem with partial conflicts

RAIRO - Operations Research, 2012

The two-dimensional bin packing problem is a well-known problem for which several exact and approximation methods were proposed. In real life applications, such as in Hazardous Material transportation, transported items may be partially incompatible, and have to be separated by a safety distance. This complication has not yet been considered in the literature. This paper introduces this extension called the two-dimensional bin packing problem with partial conflicts (2BPPC) which is a 2BP with distance constraints between given items to respect, if they are packed within a same bin. The problem is NPhard since it generalizes the BP, already NP-hard. This study presents a mathematical model, two heuristics and a multi-start genetic algorithm for this new problem.