Irregular packing problems: A review of mathematical models

Leão, Aline Aparecida de Souza; Toledo, Franklina Maria Bragion de; Oliveira, José Fernando; Carravilla, Maria Antónia; Álvarez-Valdés, Ramón

doi:10.1016/j.ejor.2019.04.045

Cited by 111 publications

(55 citation statements)

References 43 publications

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“…Extensive efforts have been devoted to solving 2D packing problems (e.g., for texture atlas refinement [LFY ∗ 19] and artistic primitive layout [RRS13]), packing similar objects on a 3D freeform surface (e.g., for architecture design [SHWP09] and personal fabrication [CML ∗ 17]), and packing regular/irregular objects within a 3D container (usually for storage and shipping). For a comprehensive review on packing problems, we refer readers to [CKPT17, LTO ∗ 20].…”

Section: Computational Analysis Of Assembliesmentioning

confidence: 99%

State of the Art on Computational Design of Assemblies with Rigid Parts

Wang

Song

Pauly

2021

Computer Graphics Forum

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An assembly refers to a collection of parts joined together to achieve a specific form and/or functionality. Designing assemblies is a non-trivial task as a slight local modification on a part's geometry or its joining method could have a global impact on the structural and/or functional performance of the whole assembly. Assemblies can be classified as structures that transmit force to carry loads and mechanisms that transfer motion and force to perform mechanical work. In this state-of-the-art report, we focus on computational design of structures with rigid parts, which generally can be formulated as a geometric modeling and optimization problem. We broadly classify existing computational design approaches, mainly from the computer graphics community, according to high-level design objectives, including fabricability, structural stability, reconfigurability, and tileability. Computational analysis of various aspects of assemblies is an integral component in these design approaches. We review different classes of computational analysis and design methods, discuss their strengths and limitations, make connections among them, and propose possible directions for future research.

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Section: Computational Analysis Of Assembliesmentioning

confidence: 99%

State of the Art on Computational Design of Assemblies with Rigid Parts

Wang

Song

Pauly

2021

Computer Graphics Forum

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“…These proposed models include the mixedinteger programming (MIP) model with a linear objective function and mixed-integer constraints [2], [15], [22], [32], [40], [42], nonlinear programming models with a nonlinear objective function [14], [15], [29], [30], [44], and constraintbased programming models where the packing problem is described by constraints [12], [38], [39]. Interested readers are referred to the references in the survey [33]. However, since the irregular packing problem is NP-hard [23], it might not be possible to compute an optimal solution within the time limit.…”

Section: Introductionmentioning

confidence: 99%

“…However, since the irregular packing problem is NP-hard [23], it might not be possible to compute an optimal solution within the time limit. For example, computing the optimal solution for 5 parts can take more than 6 hours [33]. Moreover, since these exact mathematical models usually involve a considerable number of variables and complex constraints [31], [33], generating a high-utilization solution with commercial solvers is difficult to achieve, especially for large-scale packing problems.…”

Section: Introductionmentioning

confidence: 99%

Greedy Adaptive Search: A New Approach for Large-Scale Irregular Packing Problems in the Fabric Industry

Cui

2020

IEEE Access

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The 2-dimensional irregular packing problems are important in the fabric industry. Under several restrictions, fabric packing problems require placing a given set of parts within a fixed-width rectangular sheet, aiming at a minimum length use. In textile industry production, the fabric packing problems are usually large-scale with time limits, where the total number of parts is large, and a highutilization solution should be computed in several minutes. However, there are few existing works on largescale packing problems. In this paper, we propose a greedy adaptive search algorithm by constructing a new evaluation function and introducing a new restricted local search strategy. In our algorithm, with a given initial sequence of parts, we iteratively search the best-fit part in succeeding several parts and place it on sheet. Moreover, we employ a two-stage heuristic searching algorithm to search over all the possible sequences for a good initial sequence with high utilization. Numerical examples involve some large-scale industrial instances, together with some large-scale instances generated from benchmarks. Numerical tests show that our algorithm outperforms existing state-of-the-art solvers in large-scale packing problems. The results show the potential of our algorithm to large-scale packing problems in industrial production.

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“…From mathematical point of view, this paper considers a new problem of optimal layout ellipses in a disconnected polygonal region with convex components of connectedness. This problem arises in computational geometry and operational research (Leao Aline et al 2019;Fasano 2015). In recent years the problem of optimal placement for ellipses was intensively studied by different authors (see, e.g., (Litvinchev, Infante, and Ozuna 2015a;Pankratov, Romanova, and Litvinchev 2018;Stoyan, Pankratov, and Romanova 2016;Kampas, Castillo, and Pintér 2019;Emmendorfer, Oro, and Beckel 2015;Kallrath and Rebennack 2014;Birgin, Lobato, and Martinez 2016;Pankratov, Romanova, and Litvinchev 2019) and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Optimal layout of ellipses and its application for additive manufacturing

Romanova

Stoyan

Панкратов

et al. 2019

International Journal of Production Research

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The paper studies a layout problem of variable number of ellipses with variable sizes placed into an arbitrary disconnected polygonal domain with maximum packing factor. The ellipses can be continuously translated and rotated. Restrictions on the dimensions of the ellipses are taken into account. Tools for the mathematical modeling of placement constraints (distance constraints between ellipses and containment of ellipses into a polygonal domain) using the phi-function technique are introduced. The tools make it possible to formulate the layout problem in the form of MIP model that is equivalent to a sequence of nonlinear programming subproblems. We develop a new solution algorithm that involves the feasible starting point algorithm and optimization procedure to search for efficient locally optimal solutions of the layout problem. This algorithm can be used in the design of parts for «support-free» additive manufacturing, taking into account the conditions for its static/ dynamic strength. Results of the algorithm implementation for a topologically optimized flat part with the analysis of a stress state are provided.

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Irregular packing problems: A review of mathematical models

Cited by 111 publications

References 43 publications

State of the Art on Computational Design of Assemblies with Rigid Parts

State of the Art on Computational Design of Assemblies with Rigid Parts

Greedy Adaptive Search: A New Approach for Large-Scale Irregular Packing Problems in the Fabric Industry

Optimal layout of ellipses and its application for additive manufacturing

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