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This paper disscuses the minimal area rectangular packing problem of how to pack a set of specified, non-overlapping rectangels into a rectangular container of minimal area. We investigate different mathematical programming approaches of this and introduce a novel approach based on non-linear optimization and the \\\"tunneling effect\\\" achieved by a relaxation of the non-overlapping constraints.
Background and purpose Inherently, IMRT treatment planning involves compromising between different planning goals. Multi-criteria IMRT planning directly addresses this compromising and thus makes it more systematic. Usually, several plans are computed from which the planner selects the most promising following a certain procedure. Applying Pareto navigation for this selection step simultaneously increases the variety of planning options and eases the identification of the most promising plan. Material and methods Pareto navigation is an interactive multi-criteria optimization method that consists of the two navigation mechanisms “selection” and “restriction”. The former allows the formulation of wishes whereas the latter allows the exclusion of unwanted plans. They are realized as optimization problems on the so-called plan bundle – a set constructed from precomputed plans. They can be approximately reformulated so that their solution time is a small fraction of a second. Thus, the user can be provided with immediate feedback regarding his or her decisions.