Hybrid Spatial Representation Models for Strip Packing Problems

We propose two mixed-integer linear programming based approaches for the 2-D orthogonal strip packing problem. Using the knowledge of the alternative forms of time representation in scheduling formulations, we show how to efficiently combine three different concepts into the xand ydimensions. One model features a discrete representation of the x-axis (strip width) and a continuous representation with general precedence variables in the y-axis (strip height). The other features a full continuous-space representation with the same approach for the y-axis and a single non-uniform grid made up of slots, for the x-axis. The drawback of the latter is that it requires an iterative search procedure over the number of slots to find the global optimal solution to the problem whereas the new hybrid discrete/continuous-space model needs to be solved only once. Through the solution of a set of twenty nine instances from the literature, we show that this is a better approach, even when compared to three alternative MILP models ranging from a pure discrete-space to a pure continuousspace with precedence variables in both dimensions.