A heuristic approach for packing identical rectangles in convex regions
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 , 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 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 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.,where Di0, Dj0 denote the interior of Di, Dj;
- •
the items must lie within the container, i.e.,
- •
items: variable xi, fixed ; container: fixed x0, variable this case we aim at minimizing the area (or volume) of the container, which is usually obtained as some function of the variable parameter ;
- •
items: variable xi, variable ; container: fixed x0, fixed 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 ;
- •
items: variable xi, fixed ; container: fixed x0, fixed this case we aim at maximizing the number N of items which can be placed within the container without violating the constraints.
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 () 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.,where the functions gj, j=1,…,m, are convex ones (we also assume that C is a bounded set). In the first variant [32] the position vector xi is made up by three components , where the first two identify the center of the rectangle, while 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 xi still has three components cxi, cyi, pi, 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 , and to 1 if the rectangle is in vertical position . Finally, the third variant [34] is a combination of the two previous ones, where a common angle common to all the items is used in addition to the rotations represented by the binary variables pi.
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.
Section snippets
Mathematical model of the problem
In this section we report the mathematical model proposed in [33] for the packing of equal rectangles into a convex region, when the rectangles are allowed to be either in horizontal or vertical position. Therefore, the container is identified by a finite number of convex inequalities as in (3), while each item i has fixed size parameters a, b, and variable position parameters cxi, cyi, corresponding to the center of the rectangles, and pi, a binary variable giving the orientation (horizontal,
The proposed heuristic approach
The heuristic approach proposed in this paper belongs to the class of Iterated Local Search (ILS) approaches (see, e.g., [35]), if we employ the terminology of combinatorial optimization problems, or Monotonic Basin Hopping (MBH) approaches (see, e.g., [36], [37]) if we employ the terminology of continuous optimization problems. Due to the mixed nature (discrete and continuous variables) of the problem at hand, we report both terminologies. A general scheme of the approach is depicted in
Computational experiments
For the computational experiments we considered the set of problem instances employed in [33], [34]. The details of the problem instances are reported in Table 1. These include the convex functions describing the container (column Container Data, subcolumn Constraints), the area of the container (column Container Data, subcolumn Area), the width a and height b of the rectangles (column Piece Data, subcolumn a×b), the area of the rectangles (column Piece Data, subcolumn Area), the best value N
Conclusion
In this paper we tackled the problem of packing equal rectangles within a convex region. Following [33] the problem can be reduced to the solution of mixed integer global optimization problems. We proposed a heuristic approach to solve such problems. The approach is an Iterated Local Search (or Monotonic Basin Hopping) one. The components of the heuristic have been defined and for the main one, the perturbation operation, different options, based on continuous and combinatorial moves, have been
References (38)
- et al.
Solving circle packing problems by global optimization: numerical results and industrial applications
European Journal of Operational Research
(2008) - et al.
An improved typology of cutting and packing problems
European Journal of Operational Research
(2007) - et al.
Packing equal circles in a square: a deterministic global optimization approach
Discrete Applied Mathematics
(2002) - et al.
Efficiently packing unequal disks in a circle
Operations Research Letters
(2008) - et al.
Minimizing the object dimensions in circle and sphere packing problems
Computers and Operations Research
(2008) - et al.
A mathematical model and a solution method for the problem of placing various-sized circles into a strip
European Journal of Operational Research
(2004) - et al.
An improved algorithm for the packing of unequal circles within a larger containing circle
European Journal of Operational Research
(2002) - et al.
An effective hybrid algorithm for the problem of packing circles into a larger containing circle
Computers & Operations Research
(2005) - et al.
Optimizing the packing of cylinders into a rectangular container: a nonlinear approach
European Journal of Operational Research
(2005) - et al.
New and improved results for packing identical unitary radius circles within triangles, rectangles and strips
Computers & Operations Research
(2010)
Mathematical model and solution method of optimization problem of placement of rectangles and circles taking into account special constraints
International Transactions in Operational Research
The geometry of nesting problems: a tutorial
European Journal of Operational Research
An iterated local search algorithm based on nonlinear programming for the irregular strip packing problem
Discrete Optimization
Solving irregular strip packing problems by hybridising simulated annealing and linear programming
European Journal of Operational Research
Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization
Computers & Operations Research
Orthogonal packing of rectangles within isotropic convex regions
Computers & Industrial Engineering
Disk packing in a square: a new global optimization approach
INFORMS Journal on Computing
Improving dense packings of equal disks in a square
The Electronic Journal of Combinatorics
Equal circles packing in square II: new results for up to 100 circles using the TAMSASS-PECS algorithm
Cited by (17)
A cutting plane method and a parallel algorithm for packing rectangles in a circular container
2022, European Journal of Operational ResearchCitation Excerpt :The problem of packing rectangular items into a rectangular container is the most commonly found in the literature (Boschetti, Mingozzi, & Hadjiconstantinou, 2002; Caprara & Monaci, 2004; Castro & Oliveira, 2011), and it includes many variants, such as packing with orthogonal rotation (Cintra, Miyazawa, Wakabayashi, & Xavier, 2008) and arranging items to consider guillotine cuts (Furini, Malaguti, & Thomopulos, 2016; Lodi & Monaci, 2003; Martin, Morabito, & Munari, 2020). We can also find the packing of circular items into a rectangular container (Hifi & M’hallah, 2009; Pedroso, Cunha, & Tavares, 2016), rectangular items into an arbitrary convex region (Cassioli & Locatelli, 2011), and even irregular shaped items into rectangular containers (Bennell & Oliveira, 2009). The problem of packing rectangles into a circular container is still underinvestigated, despite its importance in practice.
Orientational variable-length strip covering problem: A branch-and-price-based algorithm
2021, European Journal of Operational ResearchCitation Excerpt :Numerous studies have been conducted on the MCLP and its variants, such as those by Current and Storbeck (1988), Pirkul and Schilling (1990) and Haghani (1996), who investigate a capacitated MCLP; Kerkkamp and Aardal (2016), who research a weighted MCLP; Revelle and Hogan (1989), who examine a probabilistic MCLP; Berman and Krass (2002) and Berman, Krass, and Drezner (2003), who study a partial MCLP; and Murray and Tong (2007), who research a continuous MCLP. Another class of related problems is packing problems, which have been extensively investigated (Addis, Locatelli, & Schoen, 2008; Andrea & Marco, 2011; Côte, Gendreau, & Potvin, 2014; Erdos & Graham, 1975; Friedman, 2009; Kennedy, 2006). Most packing problems place convex or non-convex polygon objects into rectangular containers of certain lengths or widths.
Packing rectangles into a fixed size circular container: Constructive and metaheuristic search approaches
2020, European Journal of Operational ResearchCitation Excerpt :They formulated the problem as a MINLP using a branch & bound technique coupled with active set strategies. Similarly to Birgin et al. (2006b), Cassioli and Locatelli (2011) formulated the orthogonal packing of identical rectangle into a convex region as an unconstrained mixed integer global optimisation problem and solved it using an iterated local search combined with the L-BFGS algorithm of Liu and Nocedal (1989). López and Beasley (2018b) considered the orthogonal packing of unequal rectangles into a circular container of fixed size.
Goal programming application for the decision support in the daily production planning of sawmills
2019, Forest Policy and EconomicsCitation Excerpt :Exact algorithms, heuristics and metaheuristics methodologies were used in these works for solving the cutting problems. Exact algorithms only find the optimal solution for small-scale problems (Hinostroza et al., 2013; Birgin and Lobato, 2010) while heuristic (Cassioli and Locatelli, 2011; Burke et al., 2004) and metaheuristic (Martins and Tsuzuki, 2010; Talbi, 2009) techniques can address large-scale problems but optimal solutions cannot be assured. The availability of a complete and exhaustive set of CPs considering the existing technology in the industry is a key aspect to attain efficient solutions that must be incorporated in the decision making process.
Packing unequal rectangles and squares in a fixed size circular container using formulation space search
2018, Computers and Operations ResearchBoard cutting from logs: Optimal and heuristic approaches for the problem of packing rectangles in a circle
2013, International Journal of Production EconomicsCitation Excerpt :This represents a family of problems known as cutting and packing (Wäscher et al., 2007; Dyckhoff, 1990), with several particular cases belonging to the NP-hard class. Among the methods proposed to solve these problems, there is a predominance of heuristic techniques to find good solutions within reasonable computing times (Cassioli and Locatelli, 2011; Huang et al., 2007). Although two-dimensional cutting and packing problems have been extensively studied (Lodi et al., 2002), a particular case that, to the best of our knowledge, has not been dealt with in depth is the problem of packing a set of rectangles into a circular space, which arises in the cutting of boards from logs in a lumber mill.