Thiago Alves de Queiroz

Algorithms for 3D guillotine cutting problems

We present algorithms for the following three-dimensional (3D) guillotine cutting problems: unbounded knapsack, cutting stock and strip packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the unbounded 3D knapsack problem, we extend the recurrence formula proposed by [1] for the rectangular knapsack problem and present a dynamic programming algorithm that uses reduced raster points. We also consider a variant of the unbounded knapsack problem in which the cuts must be staged. For the 3D cutting stock problem and its variants in which the bins have different sizes (and the cuts must be staged), we present column generation-based algorithms. Modified versions of the algorithms for the 3D cutting stock problems with stages are then used to build algorithms for the 3D strip packing problem and its variants. The computational tests performed with the algorithms described in this paper indicate that they are useful to solve instances of moderate size.

Heuristics for two-dimensional knapsack and cutting stock problems with items of irregular shape

In this paper, the two-dimensional cutting/packing problem with items that correspond to simple polygons that may contain holes are studied in which we propose algorithms based on no-fit polygon computation. We present a GRASP based heuristic for the 0/1 version of the knapsack problem, and another heuristic for the unconstrained version of the knapsack problem. This last heuristic is divided in two steps: first it packs items in rectangles and then use the rectangles as items to be packed into the bin. We also solve the cutting stock problem with items of irregular shape, by combining this last heuristic with a column generation algorithm. The algorithms proposed found optimal solutions for several of the tested instances within a reasonable runtime. For some instances, the algorithms obtained solutions with occupancy rates above 90% with relatively fast execution time.

Heuristics for a hub location‐routing problem

We investigate a variant of the many-to-many hub location-routing problem which consists in partitioning the set of nodes of a graph into routes containing exactly one hub each, and determining an extra route interconnecting all hubs. A variable neighborhood descent with neighborhood structures based on remove/add, swap and exchange moves nested with routing and location operations is used as a local search procedure in a multistart algorithm. We also consider a sequential version of this local search in the multistart. In addition, a biased random-key genetic algorithm working with a local search routine, which also considers routing and location operations, is applied to the problem. To compare the heuristic solutions, we develop an integer programming formulation which is solved with a branch-andcut algorithm. Capacity and path elimination constraints are added in a cutting plane fashion. The separation algorithms are based on the computation of min-cut trees and on the connected components of a support graph. Computational experiments were conducted on several benchmark instances of routing problems and show that the heuristics are effective on medium to large-sized instances, while the branch-and-cut algorithm solves small to medium sized problems to optimality. These algorithms were also compared with a commercial hybrid solver showing that the heuristics are quite competitive.

Order and static stability into the strip packing problem

This paper investigates the two-dimensional strip packing problem considering the case in which items should be arranged to form a physically stable packing satisfying a predefined item unloading order from the top of the strip. The packing stability analysis is based on conditions for the static equilibrium of rigid bodies, differing from others strategies which are based on area and percentage of support. We consider an integer linear programming model for the strip packing problem with the order constraint, and a cutting plane algorithm to handle stability, leading to a branch-and-cut approach. We also present two heuristics: the first is based on a stack building algorithm; and, the last is a slight modification of the branch-and-cut approach. The computational experiments show that the branch-and-cut model can handle small and medium-sized instances, whereas the heuristics found almost optimal solutions quickly for several instances. With the combination of heuristics and the branch-and-cut algorithm, many instances are solved to near optimality in a few seconds.

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.

Two-dimensional Disjunctively Constrained Knapsack Problem: Heuristic and exact approaches

Abstract This work deals with the 0–1 knapsack problem in its two-dimensional version considering a conflict graph, where each edge in this graph represents a pair of items that must not be packed together. This problem arises as a subproblem of the bin packing problem and in supply chain scenarios. We propose some integer programming formulations that are solved with a branch-and-cut algorithm. The formulation is based on location-allocation variables mixing the one- and two-dimensional versions of this problem. When a candidate solution is found, a feasibility test is performed with a constraint programming algorithm, which verifies if it satisfies the two-dimensional packing constraints. Moreover, bounds and valid cuts are also investigated. A heuristic that generates iteratively a solution and has components of Tabu search and Simulated Annealing approaches is proposed. The results are extended to consider complete shipment of items, where subsets of items all have to be loaded or left out completely. This constraint is applied in many real-life packing problems, such as packing parts of machinery, or when delivering cargo to different customers. Experiments on several instances derived from the literature indicate the competitiveness of our algorithms, which solved 99% of the instances to optimality requiring short computational time.

Solving a Variant of the Hub Location-Routing Problem

We investigate a variant of the many-to-many (hub) location-routing problem, which consists in partitioning the set of vertices of a graph into cycles containing exactly one hub each, and determining an extra cycle interconnecting all hubs. A local search heuristic that considers add/remove and swap operations is developed.Also, a branch-and-cut approach that solves an integer formulation is investigated.Computational experiments on several instances adapted from literature show that our algorithms are good to deal with small to medium-sized instances.

On the L-approach for generating unconstrained two-dimensional non-guillotine cutting patterns

Many cutting problems on two- or three-dimensional objects require that the cuts be orthogonal and of guillotine type. However, there are applications in which the cuts must be orthogonal but need not be of guillotine type. In this paper we focus on the latter type of cuts on rectangular bins. We investigate the so-called

Dynamic cargo stability in loading and transportation of containers

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