How to Pack Trapezoids: Exact and Evolutionary Algorithms

The purposes of this paper are twofold. In the first, we describe an exact polynomial-time algorithm for the pair sequencing problem and show how this method can be used to pack fixed-height trapezoids into a single bin such that interitem wastage is minimised. We then go on to examine how this algorithm can be combined with bespoke evolutionary and local search methods for tackling the multiple-bin version of this problem—one that is closely related to one-dimensional bin packing. In the course of doing this, a number of ideas surrounding recombination, diversity, and genetic repair are also introduced and analysed.