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Zbl 1201.90167
Bennell, J.; Scheithauer, G.; Stoyan, Y.; Romanova, T.
Tools of mathematical modeling of arbitrary object packing problems. (English)
[J] Ann. Oper. Res. 179, 343-368 (2010). ISSN 0254-5330; ISSN 1572-9338/e

Summary: The article reviews the concept of and further develops phi-functions ($\Phi$-functions) as an efficient tool for mathematical modeling of two-dimensional geometric optimization problems, such as cutting and packing problems and covering problems. The properties of the phi-function technique and its relationship with Minkowski sums and the nofit polygon are discussed. We also describe the advantages of phi-functions over these approaches. A clear definition of the set of objects for which phi-functions may be derived is given and some exceptions are illustrated. A step by step procedure for deriving phi-functions illustrated with examples is provided including the case of continuous rotation.

MSC 2000:: *90C27 Combinatorial programming

Keywords: mathematical modeling; cutting and packing; phi-function; geometry; nofit polygon

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Zentralblatt MATH Berlin [Germany]