Placement Problems for Irregular Objects: Mathematical Modeling, Optimization and Applications

We describe our methodology for solving NP-hard irregular placement problems. We deal with an accurate representation of objects bounded by circular arcs and line segments and allow their free rotations within a container. We formulate a basic irregular placement problem (IRPP), which covers a wide spectrum of practical packing, cutting, nesting, clustering, and layout problems. We provide a nonlinear programming (NLP) model of the problem, employing the phi-function technique. Our model involves a large number of inequalities with nonsmooth functions. We describe a solution tree for our placement problem and evaluate the number of its terminal nodes. We reduce IRPP problem to a sequence of NLP-subproblems with smooth functions.

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