Solving the Multi-Choice Two Dimensional Shelf Strip Packing Problem with Time Windows

In the tool coating field, scheduling of production lines requires solving an optimisation problem which we call the multi-choice two-dimensional shelf strip packing problem with time windows. A set of rectangular items needs to be packed in two stages: items are placed on shelves, which in turn are placed on one of several available strips. Crucially, the item's width depends on the chosen strip and each item is associated with a time window such that items can only be placed on the same shelf if their time windows overlap. In collaboration with an industrial partner, this real-world optimisation problem is tackled in this paper by both exact and heuristic methods. The exact method is an arc-flow-based integer linear programming formulation, solved with the commercial solver CPLEX. Experimental evaluation shows that this approach can solve instances to proven optimality with up to 20 different item sizes. Larger, more realistic instances are solved heuristically by an adaptive large neighbourhood search, using first fit and best fit decreasing approaches as repair heuristics. In this way, we obtain high-quality solutions with a remaining optimality gap below 3.3% for instances with up to 2000 different item sizes. The work reported is due to be incorporated into an end-to-end decision support system with the industrial partner.