The Dotted-Board Model: A new MIP model for nesting irregular shapes

Abstract

The nesting problem, also known as irregular packing problem, belongs to the generic class of cutting and packing (C&P) problems. It di↵ers from other 2-D C&P problems in the irregular shape of the pieces. This paper proposes a new mixed-integer model in which binary decision variables are associated with each discrete point of the board (a dot) and with each piece type. It is much more flexible than previously proposed formulations and solves to optimality larger instances of the nesting problem, at the cost of having its precision dependent on board discretization. To date no results have been published concerning optimal solutions for nesting problems with more than 7 pieces. We ran computational experiments on 45 problem instances with the new model, solving to optimality 34 instances with a total number of pieces ranging from 16 to 56, depending on the number of piece types, grid resolution and the size of the board. A strong advantage of the model is its insensitivity to piece and board geometry, making it easy to extend to more complex problems such as non-convex boards, possibly with defects. Additionally, the number of binary variables does not depend on the total number of pieces but on the number of piece types, making the model particularly suitable for problems with few piece types. The discrete nature of the model requires a trade-o↵ between grid resolution and problem size, as the number of binary variables grows with the square of the selected grid resolution and with board size.

MIP models for the irregular strip packing problem: new symmetry breaking constraints

“…For example, it may have two groups of connected pieces with no interconnection. To enforce all pieces are connected, one can impose all cut sets of pieces are connected, as defined by constraints (12).…”

“…The models differ in the geometric structures used to avoid overlap among pieces. Toledo et al [12] proposed the dotted-board model where the pieces were placed in a finite set of dots inside the board. The optimality of the model is subject to the discretization used to generate the dots.…”

Nesting problems

“…We propose two mixed-integer programming models for the strip packing problem; a new dot structure to handle the geometry of cutting and packing problems; pieces rotations were included to the dotted board model (Toledo et al, 2013) and a matheuristic was build using it; constraint programming methods were proposed to all variants of cutting and packing problems classied by Wäscher et al (2007). Also, a global constraint is proposed to eliminate the overlap between pieces.…”

“…Toledo et al (2013) proposed a model where the reference point of the pieces can only be positioned at the dots of a given grid on the board. The grid used was regular, i.e., the horizontal distance between any two dierent points of the grid must be multiple of constant g x .…”

“…Toledo et al (2013) developed a mixed-integer programming model to represent the problem also considering a discrete approximation. This geometric representation is also used by heuristics (Carravilla and Ribeiro, 2005;Bennell and Dowsland, 2001;Dowsland et al, 1998).…”