Marina Andretta

Practical active-set Euclidian trust-region method with spectral projected gradients for bound-constrained minimization

A practical active-set method for bound-constrained minimization is introduced. Within the current face the classical Euclidian trust-region method is employed. Spectral projected gradient directions are used to abandon faces. Numerical results are presented.

Robust mixed-integer linear programming models for the irregular strip packing problem

Two-dimensional irregular strip packing problems are cutting and packing problems where small pieces have to be cut from a larger object, involving a non-trivial handling of geometry. Increasingly sophisticated and complex heuristic approaches have been developed to address these problems but, despite the apparently good quality of the solutions, there is no guarantee of optimality. Therefore, mixed-integer linear programming (MIP) models started to be developed. However, these models are heavily limited by the complexity of the geometry handling algorithms needed for the piece non-overlapping constraints. This led to pieces simplifications to specialize the developed mathematical models. In this paper, to overcome these limitations, two robust MIP models are proposed. In the first model (DTM) the non-overlapping constraints are stated based on direct trigonometry, while in the second model (NFP−CM) pieces are first decomposed into convex parts and then the non-overlapping constraints are written based on nofit polygons of the convex parts. Both approaches are robust in terms of the type of geometries they can address, considering any kind of non-convex polygon with or without holes. They are also simpler to implement than previous models. This simplicity allowed to consider, for the first time, a variant of the models that deals with piece rotations. Computational experiments with benchmark instances show that NFP−CM outperforms both DTM and the best exact model published in the literature. New real-world based instances with more complex geometries are proposed and used to verify the robustness of the new models.

Partial spectral projected gradient method with active-set strategy for linearly constrained optimization

A method for linearly constrained optimization which modifies and generalizes recent box-constraint optimization algorithms is introduced. The new algorithm is based on a relaxed form of Spectral Projected Gradient iterations. Intercalated with these projected steps, internal iterations restricted to faces of the polytope are performed, which enhance the efficiency of the algorithm. Convergence proofs are given and numerical experiments are included and commented. Software supporting this paper is available through the Tango Project web page: http://www.ime.usp.br/∼egbirgin/tango/.

Circle Covering Representation for Nesting problems with continuous rotations

Abstract This paper analyses distinct methods to represent a polygon through circle covering, which satisfy specific requirements, that impact primarily the feasibility and the quality of the layout of final solution. The trade-off between the quality of the polygonal representation and its derived number of circles is also discussed, showing the impact on the resolution of the problem, in terms of computational efficiency. The approach used to tackle the Nesting problem in strip packing uses a Non-Linear Programming model. Addressing these problems allows to tackle real world problems with continuous rotations.

A biased random key genetic algorithm for open dimension nesting problems using no-fit raster

Irregular 2D cutting problems with one or two open dimensions are tackled.The no-fit raster concept is extended to deal with free form items.A BRKGA combined with bottom-left heuristics is proposed to solve the problems.It outperforms recent methods from the literature on different set of instances.Instances with items as circles, convex and non-convex polygons are solved. We consider two NP-hard open dimension nesting problems for which a set of items has to be packed without overlapping into a two-dimensional bin in order to minimize one or both dimensions of this bin. These problems are faced by real-life applications, such as textile, footwear and automotive industries. Therefore, there is a need for specialized systems to help in a decision making process. Bearing this in mind, we derive new concepts as the no-fit raster, which can be used to check overlapping between any two-dimensional generic-shaped items. We also use a biased random key genetic algorithm to determine the sequence in which items are packed. Once the sequence of items is determined, we propose two heuristics based on bottom-left moves and the no-fit raster concept, which are in turn used to arrange these items into the given bin observing the objective criteria. As far as we know, the problem with two-open dimensions is being solved for the first time in the context of nesting problems and we present the first whole quadratic model for this problem. Computational experiments conducted on benchmark instances from the literature (some from the textile industry and others including circles, convex, and non-convex polygons) show the competitiveness of the approaches developed as they were able to calculate the best results for 74.14% of the instances. It can be observed that these results show new directions in terms of solving nesting problems whereby approaches can be coupled in existing intelligent systems to support decision makers in this field.

Constraint Aggregation in Non-linear Programming Models for Nesting Problems

The Nesting problem is a complex problem that arises in industries where sets of pieces or space must be efficiently placed or allocated in order to minimize wasted space or wasted raw materials, without overlaps between pieces and fully contained inside a container. This paper analyses the impact that aggregating constraints can achieve in the reduction of computational cost of a Non-Linear Programming model for Nesting problems with continuous rotations. This is achieved by aggregating non-overlapping constraints and using spatial partition and hierarchical overlap detection methods. When aggregating constraints there is also an effect of reducing the sensitivity of the solver, which may reduce the quality of the final layout. Analyzing the trade-off between constraints aggregation and the impact on the quality of the final solution is an important issue to handle nesting instances with a large number of pieces. Computational experiments show that aggregating non-overlapping constraints allows the Non-Linear Programming model for Nesting problems to scale well to tackle large size real world problems with continuous rotations.

Packing Circles and Irregular Polygons using Separation Lines

In this paper we propose a nonlinear mathematical model for the problem of packing circles, convex and nonconvex irregular polygons, within a rectangular envelope to be produced, obeying containment constraints and non-overlapping constraints; the objective of the problem is to minimize the area of the rectangular envelope. We consider free rotations of the polygons and use separation lines to ensure non-overlapping. Computational tests were run using instances presented in the literature that deal with circles and polygons simultaneously; different solutions, in which the area of the rectangular envelope is smaller than or equal to the ones found in the literature were found in most cases, and the execution time is very low. This indicates that our model is computationally efficient.

Solving Irregular Strip Packing Problems With Free Rotations Using Separation Lines

Solving nesting problems or irregular strip packing problems is to position polygons in a fixed width and unlimited length strip, obeying polygon integrity containment constraints and non-overlapping constraints, in order to minimize the used length of the strip. To ensure non-overlapping, we used separation lines. A straight line is a separation line if given two polygons, all vertices of one of the polygons are on one side of the line or on the line, and all vertices of the other polygon are on the other side of the line or on the line. Since we are considering free rotations of the polygons and separation lines, the mathematical model of the studied problem is nonlinear. Therefore, we use the nonlinear programming solver IPOPT (an algorithm of interior points type), which is part of COIN-OR. Computational tests were run using established benchmark instances and the results were compared with the ones obtained with other methodologies in the literature that use free rotation.

A general heuristic for two-dimensional nesting problems with limited-size containers

Cutting raw-material into smaller parts is a fundamental phase of many production processes. These operations originate raw-material waste that can be minimised. These problems have a strong economic and ecological impact and their proper solving is essential to many sectors of the economy, such as the textile, footwear, automotive and shipbuilding industries, to mention only a few. Two-dimensional (2D) nesting problems, in particular, deal with the cutting of irregularly shaped pieces from a set of larger containers, so that either the waste is minimised or the value of the pieces actually cut from the containers is maximised. Despite the real-world practical relevance of these problems, very few approaches have been proposed capable of dealing with concrete characteristics that arise in practice. In this paper, we propose a new general heuristic (H4NP) for all 2D nesting problems with limited-size containers: the Placement problem, the Knapsack problem, the Cutting Stock problem, and the Bin Packing problem. Extensive computational experiments were run on a total of 1100 instances. H4NP obtained equal or better solutions for 73% of the instances for which there were previous results against which to compare, and new benchmarks are proposed.

Deterministic and stochastic global optimization techniques for planar covering with ellipses problems

Problems of planar covering with ellipses are tackled in this work. Ellipses can have a fixed angle or each of them can be freely rotated. Deterministic global optimization methods are developed for both cases, while a stochastic version of the method is also proposed for large instances of the latter case. Numerical results show the effectiveness and efficiency of the proposed methods.