A heuristic approach for packing identical rectangles in convex regions

Abstract

In this paper we propose a heuristic approach for the problem of packing equal rectangles within a convex region. The approach is based on an Iterated Local Search scheme, in which the key step is the perturbation move. Different perturbation moves, both combinatorial and continuous ones, are proposed and compared through extensive computational experiments on a set of test instances. The overall results are quite encouraging.

Introduction

In packing problems some items have to be placed into some containers in such a way that the overall unused space within the container (the so called waste) is minimized. The items and the containers have a fixed (usually two- or three-dimensional) shape, while their dimensions and positions can vary. More formally, let us consider N items. For each item i we introduce the parameter vectors ${\mathbf{x}}_i,{\mathit{\alpha }}_i$, the former being usually a vector of variables, while the latter might be a vector of fixed or variable values, depending on the applications (see below). These two vectors are, respectively, a position and a size vector, which allow to uniquely identify the portion of the space where the item lies, denoted by

$$D_i=D_i({\mathbf{x}}_i,{\mathit{\alpha }}_i),i=1,\dots ,N\text{.}$$

For instance, if the items are circles, we have a two-dimensional position vector corresponding to the coordinates of the center of the circle, while we have a single size parameter, corresponding to the radius of the circle. Although the shapes of the items might be different from each other, in most cases the shape is the same for all of them. Possible shapes are convex ones such as circles, squares, rectangles, but also nonconvex ones are sometimes considered.

Next, we consider a container

$$C=C({\mathbf{x}}_0,{\mathit{\alpha }}_0)\text{,}$$

with some fixed shape (not necessarily equal to those of the items), and also depending on a position and a size vector. In some applications the vectors, identifying the portion of the space occupied by the container, are replaced by a description of such portion through proper inequalities and/or equalities.

A packing problem can be formulated as an optimization problem. The constraints of the problem are the following:

• the items may touch each other but cannot overlap, i.e.,

  $$D_i^0({\mathbf{x}}_i,{\mathit{\alpha }}_i)\cap D_j^0({\mathbf{x}}_j,{\mathit{\alpha }}_j)=\varnothing \forall i\ne j\text{,}$$

  where D_i⁰, D_j⁰ denote the interior of D_i, D_j;

• the items must lie within the container, i.e.,

  $$D_i({\mathbf{x}}_i,{\mathit{\alpha }}_i)\subseteq C({\mathbf{x}}_0,{\mathit{\alpha }}_0)\forall i\text{.}$$

The decision variables depend on the problem at hand. Usually, the position vectors x_i are variables, while the size vectors ${\mathit{\alpha }}_i$ can either be variables or fixed values. The position vector x₀ for the container is usually fixed, while its size vector may either be variable or fixed. The objective of the problem should state the fact that we aim at minimizing the waste. This can be accomplished in different ways depending on the nature of the position and size vectors:

• items: variable x_i, fixed ${\mathit{\alpha }}_i$; container: fixed x₀, variable ${\mathit{\alpha }}_0\text{—in}$ this case we aim at minimizing the area (or volume) of the container, which is usually obtained as some function of the variable parameter ${\mathit{\alpha }}_0$;

• items: variable x_i, variable ${\mathit{\alpha }}_i$; container: fixed x₀, fixed ${\mathit{\alpha }}_0\text{—in}$ this case we aim at maximizing the sum of the areas (or volumes) of the items, where the area of each item i is usually obtained as some function of the variable parameter ${\mathit{\alpha }}_i$;

• items: variable x_i, fixed ${\mathit{\alpha }}_i$; container: fixed x₀, fixed ${\mathit{\alpha }}_0\text{—in}$ this case we aim at maximizing the number N of items which can be placed within the container without violating the constraints.

Many packing problems have been tackled in the literature. Most papers present heuristic approaches, although some exact approaches have also been proposed. A detailed survey about methods and applications of packing problems can be found in [1], while a problem typology is extensively presented in [2].

Probably, the most widely studied cases are those involving circular items within containers having some regular form such as a circle or a square. In the field of circular items/square container we recall some heuristic (see, e.g., [3], [4], [5], [6], [7]) and exact approaches (see, e.g., [8], [9], [10], [11]). We also refer to survey [12] and a book [13]. Heuristic approaches for the case of circular items/circular container include those discussed in [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. In [25] the problem of packing cylinders into a rectangular container is considered, while in [26] circular items are packed into triangles, rectangles and strips. In [27] the problem of placing circles with different sizes into a rectangular container with fixed width and minimum height is considered. An even more general problem is tackled in [28], aiming at cutting several different shapes (circles and convex polygons) from rectangular plates of raw material. Benchmark results for the problem of packing equal circles in a container whose shape is a square, a circle or an equilateral triangle are reported and continuously updated in E. Specht's web site.¹ Test instances for the problem of packing unequal circles in a circle include those in [17] and the quite challenging instances from the Circle Packing Contest.² The problem of packing irregular polygons has also been well studied (see [29] for an overview), considering either fixed size containers or infinite length strips (the so–called Irregular Strip Packing Problem). The latter has been tackled, for instance, in [30] using a local search-based method, while in [31] linear programming relaxations are solved in a Simulated-Annealing framework.

In this paper we deal with the case of rectangular items/convex container. In a series of papers [32], [33], [34] different variants of this problem have been considered. In all variants the size parameters of the rectangles, a and b ($a\ge b$) denoting the length of the two edges of the rectangles, are fixed and equal for all the rectangles. Moreover, the portion of the space occupied by the convex container is fixed and described by (convex) inequalities, i.e.,

$$C=\{(y_1,y_2)\in {\mathbb{R}}^2:g_j(y_1,y_2)\le 0,j=1,\dots ,m\}\text{,}$$

where the functions g_j, j=1,…,m, are convex ones (we also assume that C is a bounded set). In the first variant [32] the position vector x_i is made up by three components $c_x^i,c_y^i,{\theta }_i$, where the first two identify the center of the rectangle, while ${\theta }_i$ denotes the rotation of the rectangle with respect to its horizontal position (the one where the basis of the rectangle is its longest edge). In the second variant [33] the position vector x_i still has three components c_xⁱ, c_yⁱ, p_i, where the first two are as before, while the third one is a binary variable equal to 0 if the rectangle is in horizontal position $({\theta }_i=0)$, and to 1 if the rectangle is in vertical position $({\theta }_i={90}^{\circ })$. Finally, the third variant [34] is a combination of the two previous ones, where a common angle $\theta$ common to all the items is used in addition to the ${90}^{\circ }$ rotations represented by the binary variables p_i.

In this paper we will explore the second variant, although the proposed technique can be extended also to the other two variants. The paper is structured as follows. In Section 2 we will report the model for the problem proposed in [33]. In Section 3 the main components of the proposed heuristic approach are presented and discussed. Extensive computational experiments and their results are presented in Section 4.