Two-dimensional packing problems: A survey

A multi-objective ant colony system algorithm for virtual machine placement in cloud computing

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 not 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.

Exact Solution of the Two-Dimensional Finite Bon Packing Problem

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.

Heuristic and Metaheuristic Approaches for a Class of Two-Dimensional Bin Packing Problems

The scheduling of computer and manufacturing systems has been the subject of extensive research for over forty years. In addition to computers and manufacturing, scheduling theory can be applied to many areas including agriculture, hospitals and transport. The main focus is on the efficient allocation of one or more resources to activities over time. Adopting manufacturing terminology, a job consists of one or more activities, and a machine is a resource that can perform at most one activity at a time. We concentrate on deterministic machine scheduling for which it is assumed that all data that define a problem instance are known with certainty.

A Review of Machine Scheduling: Complexity, Algorithms and Approximability

Bin packing approximation algorithms: Survey and classification

The problem of on-line scheduling a set of independent jobs on m machines is considered. The goal is to minimize the makespan of the schedule. Graham’s List Scheduling heuristic [R. L. Graham, SIAM J. Appl. Math., 17(1969), pp. 416–429] guarantees a worst case performance of $2 - \frac{1} {m}$ for this problem. This worst case bound cannot be improved for $m = 2$ and $m = 3$. For $m \geqslant 4$, approximation algorithms with worst case performance at most $2 - \frac{1}{m} - \varepsilon _m $ are presented, where $\varepsilon _m $ is some positive real depending only on m.

An on-line scheduling heuristic with better worst case ratio than Graham's list scheduling

New lower bounds for certain classes of bin packing algorithms

In the classical version of the bin packing problem one is given a list L = (a 1,...,a n ) of items (or elements) and an infinite supply of bins with capacity C. A function s(a i ) gives the size of item a i , and satisfies 0 < s(a i )≤C, 1 ≤ i ≤ n. The problem is to pack the items into a minimum number of bins under the constraint that the sum of the sizes of the items in each bin is no greater than C. In simpler terms, a set of numbers is to be partitioned into a minimum number of blocks subject to a sum constraint common to each block. We use the bin packing terminology, as it eases considerably the problem of describing and analyzing algorithms.

Bin Packing Approximation Algorithms: Combinatorial Analysis

In the classical bin packing problem, one is asked to pack items of various sizes into the minimum number of equal-sized bins. In the on-line version of this problem, the packer is given the items one by one and must immediately and irrevocably assign every item to its bin, without knowing the future items. Beginning with the first results in the early 1970's, we survey — from the worst case point of view — the approximation results obtained for on-line bin packing, higher dimensional versions of the problem, lower bounds on worst case ratios and related results.

Gábor Galambos

Papers

Bin packing approximation algorithms: Survey and classification

An on-line scheduling heuristic with better worst case ratio than Graham's list scheduling

New lower bounds for certain classes of bin packing algorithms

Bin Packing Approximation Algorithms: Combinatorial Analysis

On-line bin packing ― a restricted survey