Abstract
The visualization of conveyor systems in the sense of a connected graph is a challenging problem. Starting from communication data provided by the IT system, graph drawing techniques are applied to generate an appealing layout of the conveyor system. From a mathematical point of view, the key idea is to use the concept of stress majorization to minimize a stress function over the positions of the nodes in the graph. Different to the already existing literature, we have to take care of special features inspired by the real-world problems.
Highlights
Graphs are widely used in various domains to visualize important information clearly
Drawing the physical structure of a network as a graph is an approach mainly used in the area of sensor networks [18, 25] and molecular structures [1]; to the best of our knowledge, it has never been applied to conveyor systems
[30] Zheng et al use a modified stochastic gradient descent to minimize the stress function instead of majoriztaion. This stochastic gradient descent seems to be beneficial regarding performance issues, we focus on the stress majorization proposed by Gansner, Koren and North [16] as our major concern is the generation of an appealing layout of the real conveyor system
Summary
Graphs are widely used in various domains to visualize important information clearly. Gansner et al [16] improved this algorithm by applying the majorization method for the area of multidimensional scaling to minimize the energy or stress function. We will discuss a second method for graph drawing, namely, the classic multidimensional scaling [4,5,6, 27] that computes a good initial layout for stress majorization. To obtain a good visualization of the graph, we minimize the following stress or cost function:. Since Laplacian Lω is positive definite, the global minimum is unique, and we can either use direct methods such as the Cholesky decomposition or iterative methods like conjugate gradients to solve the linear system (2.7). Input: Matrix d ∈ Rn×n with the respective distances Output: Matrix X ∈ Rn×2 with the coordinates of all nodes
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