A semi-continuous MIP model for the irregular strip packing problem

Abstract

Solving nesting problems involves the waste minimisation in cutting processes, and therefore it is not only economically relevant for many industries but has also an important environmental impact, as the raw materials that are cut are usually a natural resource. However, very few exact approaches have been proposed in the literature for the nesting problem (also known as irregular packing problem), and the majority of the known approaches are heuristic algorithms, leading to suboptimal solutions. The few mathematical programming models known for this problem can be divided into discrete and continuous models, based on how the placement coordinates of the pieces to be cut are dealt with. In this paper, we propose an innovative semi-continuous mixed-integer programming model for two-dimensional cutting and packing problems with irregular shaped pieces. The model aims to exploit the advantages of the two previous classes of approaches and discretises the y-axis while keeping the x-coordinate continuous. The board can therefore be seen as a set of stripes. Computational results show that the model, when solved by a commercial solver, can deal with large problems and determine the optimal solution for smaller instances, but as it happens with discrete models, the optimal solution value depends on the discretisation step that is used.

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

“…The optimality of the model is subject to the discretization used to generate the dots. In order to combine the accuracy of continuous models with the advantages of the dotted-board model, Leao et al [8] proposed a semi-continuous model where the pieces can be placed in continuous positions in x-axis and in discrete positions in y-axis.…”

A Family of Heuristic-Based Inequalities for Maximizing Overall Safety Margins in Aircraft Parking Stands Arrangement Problems

“…The item cutting and packing problem in twodimensional space has been widely studied in the literature because of its practical use in various industries, and such packing problems can be classified as either regular items packing problems or irregular items packing problems [15,[17][18][19][20][21][22][23][24]. Many mathematical formulations solving irregular items packing problems have been proposed in the literature [25][26][27][28][29][30]. The No-Fit Polygon (NFP) has been widely used in detecting if two irregular items overlap with each other [31,32], while other approaches also exist [33].…”

Empacotamento de itens irregulares considerando balanceamento da carga

“…Os resultados computacionais mostram que o modelo é capaz de provar a otimalidade para instâncias bem maiores do que os outros modelos, mas as soluções são dependentes da malha adotada. Leão et al (2016) propõem um modelo diferente dos modelos anteriores, por utilizar uma abordagem semicontínua para o recipiente. Assim, o recipiente, os NFPs e IFRs são definidos por conjuntos de linhas horizontais e o ponto de referência dos itens devem ser alocados nessas linhas.…”