Packing problems on generalised regular grid: Levels of abstraction using integer linear programming

IF 2.5 4区计算机科学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Graphical Models Pub Date : 2023-10-07 DOI:10.1016/j.gmod.2023.101205

Hao Hua , Benjamin Dillenburger

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Abstract

Packing a designated set of shapes on a regular grid is an important class of operations research problems that has been intensively studied for more than six decades. Representing a $d$ -dimensional discrete grid as $Z^{d}$ , we formalise the generalised regular grid (GRG) as a surjective function from $Z^{d}$ to a geometric tessellation in a physical space, for example, the cube coordinates of a hexagonal grid or a quasilattice. This study employs 0-1 integer linear programming (ILP) to formulate the polyomino tiling problem with adjacency constraints. Rotation & reflection invariance in adjacency are considered. We separate the formal ILP from the topology & geometry of various grids, such as Ammann-Beenker tiling, Penrose tiling and periodic hypercube. Based on cutting-edge solvers, we reveal an intuitive correspondence between the integer program (a pattern of algebraic rules) and the computer codes. Models of packing problems in the GRG have wide applications in production system, facility layout planning, and architectural design. Two applications in planning high-rise residential apartments are illustrated.

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广义正则网格上的填充问题:用整数线性规划的抽象层次

在规则网格上填充一组指定的形状是一类重要的运筹学问题，已经被深入研究了60多年。将d维离散网格表示为Zd，我们将广义规则网格(GRG)形式化为从Zd到物理空间中的几何镶嵌的满射函数，例如，六边形网格或准格的立方体坐标。本研究采用0-1整数线性规划(ILP)来表述具有邻接约束的多集平铺问题。旋转,考虑了邻接中的反射不变性。我们将形式ILP从拓扑中分离出来;各种网格的几何结构，如Ammann-Beenker平铺、Penrose平铺和周期超立方体。基于先进的求解器，我们揭示了整数程序(代数规则模式)与计算机代码之间的直观对应关系。GRG中的包装问题模型在生产系统、设施布局规划和建筑设计中有着广泛的应用。举例说明了在高层住宅公寓规划中的两种应用。

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来源期刊

Graphical Models 工程技术-计算机：软件工程

CiteScore

3.60

自引率

5.90%

发文量

审稿时长

47 days

期刊介绍： Graphical Models is recognized internationally as a highly rated, top tier journal and is focused on the creation, geometric processing, animation, and visualization of graphical models and on their applications in engineering, science, culture, and entertainment. GMOD provides its readers with thoroughly reviewed and carefully selected papers that disseminate exciting innovations, that teach rigorous theoretical foundations, that propose robust and efficient solutions, or that describe ambitious systems or applications in a variety of topics. We invite papers in five categories: research (contributions of novel theoretical or practical approaches or solutions), survey (opinionated views of the state-of-the-art and challenges in a specific topic), system (the architecture and implementation details of an innovative architecture for a complete system that supports model/animation design, acquisition, analysis, visualization?), application (description of a novel application of know techniques and evaluation of its impact), or lecture (an elegant and inspiring perspective on previously published results that clarifies them and teaches them in a new way). GMOD offers its authors an accelerated review, feedback from experts in the field, immediate online publication of accepted papers, no restriction on color and length (when justified by the content) in the online version, and a broad promotion of published papers. A prestigious group of editors selected from among the premier international researchers in their fields oversees the review process.