Two Dimensional Strip Packing with Unloading Constraints

New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems

Approximation and online algorithms for multidimensional bin packing: A survey

In the strip packing problem (a standard version of the two-dimensional cutting stock problem), the goal is to pack a given set of rectangles into a vertical strip of unit width so as to minimize the total height of the strip needed. The k-stage Guillotine packings form a particularly simple and attractive family of feasible solutions for strip packing. We present a complete analysis of the quality of k-stage Guillotine strip packings versus globally optimal packings: k=2 stages cannot guarantee any bounded asymptotic performance ratio. k=3 stages lead to asymptotic performance ratios arbitrarily close to 1.69103; this bound is tight. Finally, k=4 stages yield asymptotic performance ratios arbitrarily close to 1.

This paper presents a solution of a constrained two dimensional strip packing problem using genetic algorithms. The constraint consists of considering three-stage guillotine patterns. This is quite a real constraint motivated by technological considerations in some industries. An analysis of including distributed population ideas and parallelism into the basic genetic algorithm is carried out to solve the problem accurately and efficiently. Experimental evidence in this work shows that the proposed parallel versions of the distributed algorithms outperform their sequential counterparts in time, although there are no significant differences either in the mean best values obtained or in the effort.

Shelf algorithms for on-line strip packing

In the three-dimensional (3D) strip packing problem, we are given a set of 3D rectangular items and a 3D box $B$. The goal is to pack all the items in $B$ such that the height of the packing is minimized. We consider the most basic version of the problem, where the items must be packed with their edges parallel to the edges of $B$ and cannot be rotated. Building upon Caprara's work for the two-dimensional (2D) bin packing problem, we obtain an algorithm that, given any $\epsilon>0$, achieves an approximation of $T_{\infty}+\epsilon\approx1.69103+\epsilon$, where $T_{\infty}$ is the well-known number that occurs naturally in the context of bin packing. Our key idea is to establish a connection between bin packing solutions for an arbitrary instance $I$ and the strip packing solutions for the corresponding instance obtained from $I$ by applying the harmonic transformation to certain dimensions. Based on this connection, we also give a simple alternate proof of the $T_{\infty}+\epsilon$ approximation for 2D bin packing due to Caprara. In particular, we show how his result follows from a simple modification of the asymptotic approximation scheme for 2D strip packing due to Kenyon and Rémila.

We study the well-known two-dimensional strip packing problem. Given is a set of rectangular axis-parallel items and a strip of width W with infinite height. The objective is to find a packing of these items into the strip, which minimizes the packing height. Lately, it has been shown that the lower bound of 3/2 of the absolute approximation ratio can be beaten when we allow a pseudo-polynomial running-time of type \((n W)^{f(1/\varepsilon )}\), for any function f. If W is polynomially bounded by the number of items, this is a polynomial running-time. We present a pseudo-polynomial algorithm with approximation ratio \(4/3 +\varepsilon \) and running time \((n W)^{1/\varepsilon ^{\mathcal {O}(2^{1/\varepsilon })}}\).

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axis-parallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme ($\mathcal{PTAS}$) for strip packing where rotations by 90 degrees are permitted and an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided $\mathcal{P} \not= \mathcal{NP}$.

A reduction approach for solving the rectangle packing area minimization problem

\nStrip packing is a well-known NP-hard problem and it was widely applied in engineering fields. This paper considers a two-dimensional orthogonal strip packing problem. Until now some exact algorithm and mainly heuristics were proposed for two-dimensional orthogonal strip packing problem. While this paper proposes a two-stage hybrid algorithm for it. In the first stage, a heuristic algorithm based on layering idea is developed to construct a solution. In the second stage, a great deluge algorithm is used to further search a better solution. Computational results on several classes of benchmark problems have revealed that the hybrid algorithm improves the results of layer-heuristic, and can compete with other heuristics from the literature.\n

Given a set of rectangular pieces and a container of fixed width and variable height, the 2D rectangular strip packing problem consists of orthogonally placing all the pieces within the container, without overlapping, such that the overall height of the layout is minimized. It has many industrial applications such as wood or glass cutting, very large scale integration design, etc. To solve this problem, many algorithms based on different strategies have been proposed. According to the wisdom and experience of human being, a quasi-human heuristic algorithm for the 2D rectangular strip packing problem is recommended based on the existing research, some important packing strategies are presented in this paper. Twenty-one rectangle-packing, 13 random strip-packing and 10 classes strip-packing instances are used to test the performance of the algorithm developed. All of 21 rectangle-packing instances and three of 13 strip-packing instances are achieved optimal solutions, and the average relative distance of 10 classes strip-packing instances obtained by the developed algorithm is 2.28%. Experimental results demonstrate that the algorithm developed is fairly efficient in solving the 2D rectangular strip packing problem.

We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P!=NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a (7/5+epsilon)-approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm (though a PPT approximation scheme might exist). In this paper we make further progress on the PPT approximability of strip packing, by presenting a (4/3+epsilon)-approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the empty space left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by 90 degrees, achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3/2 for the case with rotations as well.

In this dissertation, we solve a school bus routing problem of transporting students including special education (handicapped) students and assigning them in Oldham county education district. The main contribution of this research is that we consider special education students (Type-2) along with other students (Type-1) and design a comprehensive school bus schedule to transport both kinds of students at the same time. Also, a student assignment mathematical model is presented to optimize the number of bus stops in use as well as one important measure of service quality, the total student walking distance. Comparing to the classic clustering methods, heuristic methods, or other methods from previous literatures, a mathematical optimization model is developed to solve a student assignment problem and to obtain the global optimal solution. The modeling constraints include budget limit, travel time limit, equity, school time window, and etc. Especially, the main difference between our model and other models is that it takes Type-2 students into consideration along with critical constraints accordingly, and solves the resulting more complex problem. Moreover, the school bus routing model in this work is one of the most general optimization models representing the school bus routing problem. On the other hand, similar to all existing models, the developed model considers the total system cost as the objective function value to minimize, different bus capacities, and common vehicle routing constraints such as flow conservation on routes and subtour elimination. Furthermore, another main difference is that the bus scheduling and school time window is also considered and solved in the model. With two different types of students, both Type-1 and Type-2, the time restrictions are varying, resulting in more complexity and additional constraints. The results in this work present the difficulties of meeting the requirement of Type-2 student riding time limit and school time window simultaneously. Also, the constraints regarding service equity and quality are provided and they can be used by decision makers if necessary. Either densely populated urban areas or sparsely populated rural areas, the school bus routing problem is difficult to solve due to a large number of students or long travel distance. The school bus routing problem falls under vehicle routing problem (VRP) with additional requirements because each student represents one unit of capacity. In this dissertation, we present a modeling framework that solves a student assignment problem with bus stop selection, and subsequently a school bus routing problem with school

We consider two- and three-dimensional bin-packing problems where 90° rotations are allowed. We improve all known asymptotic performance bounds for these problems. In particular, we show how to combine ideas from strip packing and two-dimensional bin packing to give a new algorithm for the three-dimensional strip packing problem where boxes can only be rotated sideways. We propose to call this problem “This side up”. Our algorithm has an asymptotic performance bound of 9/4.

Most research on Strip Packing Problems is focused on the single-objective formulation of the problem. However, in this work we deal with a more general and practical variant of the problem, which not only seeks to optimize the usage of the raw material, but also the production process. For the problem solution, we have applied some of the most-known multi-objective evolutionary algorithms, since they have shown a promising behavior when affording multi-objective real-world problems. For an initial implementation, we proposed a solution codification which is based on a complete representation of the pattern layouts. Such an approach was promising but wasn't suitable to afford large instances. For this reason, we have focused on the design of a codification which can be much more competitive when compared to some tailor-made methods. In this sense, we present a hyperheuristic-based codification as an alternative to combine heuristics in such a way that a heuristic's strengths make up for the drawbacks of another. Results demonstrate the advantage of using multi-objective approaches, hyperheuristic-based representations, and of course, the importance on the selection of appropriate solution codifications.