We show that a modification of the Kenyon-Remila algorithm for the strip-packing problem yields an improved bound on the value of the approximate solution. As a corollary we derive that there exists a polynomial-time algorithm that always finds a solution of value OPT +O( √ OPT logOPT ) where OPT is the optimal value.

We examine various strategies for exact approaches to the 2dimensional strip packing problem (2SP) with and without rotations of 90 degrees. We first develop a branch-and-bound algorithm based on the sequence pair representation. Next, we focus on the perfect packing problem (PP), which is a special case of 2SP where all given rectangles are required to be packed without wasted space, and desig...

In this paper a hybrid algorithm to solve Irregular Strip Packing problems is presented. The metaheuristic simulated annealing is used to guide the search over the solution space while linear programming models are solved to generate neighbourhoods during the search process. These linear programming models, which are used to locally optimise the layouts, derive from the application of compactio...

A new meta-heuristic algorithm to find the minimum height for twodimensional strip rectangular packing problem is presented. This algorithm is mainly based on the heuristic recursive strategy and simulated annealing algorithm. The computational results on a class of benchmark problems have shown that this algorithm not only finds shorter height than known meta-heuristic but also runs in shorter...

We present approximation algorithms for the three-dimensional strip packing problem, and the three-dimensional bin packing problem. We consider orthogonal packings where ninety-degree rotations are allowed. The algorithms we show for these problems have asymptotic performance bounds 2.64, and 4.89, respectively. These algorithms are for the more general case in which the bounded dimensions of t...

The problem of rectangular objects packing into a semi-endless strip is under consideration. Here we propose to solve this problem by linear cutting approximation that may be presented as block structure of packing. Sometimes it becomes necessary to rearrange elements inside linear cutting blocks. The determined method of “reconstruction”(REC) proposed below helps to get solutions close to opti...

We consider two-dimensional bin packing and strip packing problems where the items have to be packed by levels. We introduce new mathematical models involving a polynomial number of variables and constraints, and show that their LP relaxations dominate the standard area relaxations. We then propose new (combinatorial) bounds that can be computed in O(n log n) time. We show that they dominate th...

We consider three connected resource-constrained optimization problems: 1D bin packing (BPP), resource-constrained project scheduling (PSP), and 2D strip packing without rotation (SPP). In the context of SPP, BPP is also known as the 1D bar relaxation and PSP as the 1D contiguous relaxation. It is easy to see that SPP can be relaxed in two different ways down to a PSP: horizontal and vertical s...

In this paper, we show how the introduction of a new primitive constraint over finite domains in the constraint logic programming system CHIP allows us to find very good solutions for a large class of very difficult scheduling and placement problems. Examples on the cumulative scheduling problem, the 10 jobs x 10 machines problem, the perfect square problem, the strip packing problem and the in...

We consider a natural generalization of multiprocessor scheduling, the bin scheduling problem. Here, we model the situation in which working resources are available during separate time windows, or shifts, only. Restricting to scheduling without job preemption, every task must be processed completely during one shift. Our aim is to find a feasible schedule in which the number of required shifts...