A lookahead matheuristic for the unweighed variable-sized two-dimensional bin packing problem

The unweighed oriented variable-sized two-dimensional guillotine bin packing problem consists in packing without overlap small rectangular items into large non-identical rectangular bins, with the items obtained via guillotine cuts. It minimizes the waste of the used bins. It is herein approximately solved using a hybrid matheuristic, which applies a sequence of low-level mixed-integer programs that reserve space for unpacked items and that are guided by feasibility constraints and by upper bounds on the objective function. The embedded constraints constitute a lookahead mechanism that prohibits the investigation of infeasible directions and constrains the search to improving ones. The matheuristic further employs high-level diversification and intensification mechanisms. The diversification incorporates a sequential value correction algorithm that tags a pseudo-price to each item to govern the fitness functions of mixed integer programs and subsequently their solution construction process. The intensification is a local search that investigates the neighbourhood of promising solutions. The extensive computational experiments provide evidence of the good performance of the proposed matheuristic. For the variable-sized bin packing benchmark instances, the matheuristic matches and improves 90.8% of the upper bounds. For the single bin-size bin packing benchmark instances, the matheuristic further proves the optimality of 82.6% upper bounds while it matches 14.4% and improves 2.6% existing bounds.