Abstract

Abstract  Packing problems are a class of problems that seek to optimise the arrangement of objects according to a given objective while respecting a set of geometric constraints. These problems have many and diverse applications. Packing problems are NP-hard, and as a result, solution methodologies generally employ heuristics. The phi-function technique is a powerful tool to help solve packing problems by providing an analytical description of the placement constraints. It deals with arbitrary shaped 2D and 3D objects, which can be continuously rotated and translated. Using phi-functions reduces the packing problem to a Nonlinear Programming or Mixed Integer Nonlinear Programming model.  The paper will introduce the definition of a continuous and everywhere defined phi-function as well as its basic features. It will explain the concept of quasi phi-functions, which generalizes the concept of phi-functions to include auxiliary variables. This modification allows for the efficient modelling of relationships between objects that were not readily available with phi-functions. Mathematical models, solution strategies and computational results are provided for some packing problems that arise in different applications, in particular, in material sciences and additive manufacturing. The discussion will identify some future research areas.  Authors:   Prof. Julia Bennell1   1University of Leeds, Leeds University Business School    J.Bennell@leeds.ac.uk  Prof. Tetyana Romanova (speaker) 1,2  1University of Leeds, Leeds University Business School     2Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine  T.Romanova@leeds.ac.uk