Mixed integer quadratically-constrained programming model to solve the irregular strip packing problem with continuous rotations

Abstract

The irregular strip packing problem consists of cutting a set of convex and nonconvex two-dimensional polygonal pieces from a board with a fixed height and infinite length. Owing to the importance of this problem, a large number of mathematical models and solution methods have been proposed. However, only few papers consider that the pieces can be rotated at any angle in order to reduce the board length used. Furthermore, the solution methods proposed in the literature are mostly heuristic. This paper proposes a novel mixed integer quadratically-constrained programming model for the irregular strip packing problem considering continuous rotations for the pieces. In the model, the pieces are allocated on the board using a reference point and its allocation is given by the translation and rotation of the pieces. To reduce the number of symmetric solutions for the model, sets of symmetry-breaking constraints are proposed. Computational experiments were performed on the model with and without symmetry-breaking constraints, showing that symmetry elimination improves the quality of solutions found by the solution methods. Tests were performed with instances from the literature. For two instances, it was possible to compare the solutions with a previous model from the literature and show that the proposed model is able to obtain numerically accurate solutions in competitive computational times.

Two-dimensional irregular packing problems: A review

“…The MIP and LP methods solve the problem by establishing an exact mathematical model of the packing process under constraints that do not allow overlaps between parts, and the parts must be included in the master surface. The idea of solving 2D irregular layout problems by MIP and LP methods can be referred to literature (Gomes and Oliveira, 2006;Fischetti and Luzzi, 2009;Toledo et al, 2013;Santoro and Lemos, 2015;Cherri et al, 2016;Leao et al, 2016;Rodrigues and Toledo, 2017;Cherri et al, 2018). Mixed integer programming (MIP) and other operational research methods were often combined with the NFP method for 2D layout problems (Silva et al, 2010).…”

“…Liu et al (Liu et al, 2015) proposed a point-to-point interactive collision algorithm to determine the intersection and distance along the collision direction. Cherri (Cherri et al, 2018) and Peralta (Peralta et al, 2018) noticed the relationship between a point and a line, and used D-Function to calculate the distance between line and point to measure overlapping.…”

A new mixed-integer programming model for irregular strip packing based on vertical slices with a reproducible survey

“…Second, the family of Constraint Programming (CP) models based on the NFP, whose pioneering work is introduced by Ribeiro et al [69], and subsequently improved by Carravilla et al [17], Ribeiro and Carravilla [68], and Cherri et al [21]. And third, Other models based on alternatives geometric representations and Non-Linear Programming (NLP) models, such as (3.a) the family of models based on Φ-functions, whose pioneering works are introduced by Stoyan et al [88], Chernov et al [19], and Stoyan et al [89]; (3.b) others non-linear models based on direct trigonometry introduced by Rocha et al [71], Cherri et al [24], and Peralta et al [64]; and finally, (3.c) the family of models based on circle coverings admitting free rotations, whose pioneering work is introduced by Jones [48] and subsequently refined by Rocha et al [72], Rocha et al [71], and Wang et al [92].…”

“…The basic NFP-CM-VS model is defined by the objective function (17) and the constraints (18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32), whilst the full NFP-CM-VS model is defined by the former objective function and the constraints, symmetry breakings, valid inequalities, and variable eliminations in expressions . The objective function (17) together with the constraints (18)(19)(20)(21)(22)(23) fit the definition of the basic NFP-CM model without cuts [25], with the only exception of our two distinguished binary variables and encoding the left and right feasible sub-regions shown in figure 5, whilst the constraints (24-29) encode our new convex decomposition based on vertical slices, which removes all symmetric solutions derived from any relative placement between pieces.…”

“…The constraints ( 42) and ( 43) introduce two significant improvements on the basic NFP-CM-VS model as follows. First, they remove the big-M factors from constraints (24)(25)(26)(27)(28)(29). And second, they allow a significant reduction in the complexity of the model by factorizing all constraints encoding the vertical lines bounding the feasible sub-regions of each convex part into only two constraints, whilst they hold the symmetry-breaking of the symmetric solutions for the space of feasible relative placements between pieces with much fewer constraints per convex part than the state-of-the-art Improved NFP-CM model [73].…”