The Orthogonal Packing and Scheduling Problem: Model, Heuristic, and Benchmark

This paper addresses a new orthogonal packing and time scheduling problem—3D space-time optimization problem (3D-STO). Given a rectangular sheet and a set of rectangular items, each item needs to be continuously processed with a time length on the sheet, the 3D-STO consists in arranging each item’s loading time, location, and orientation for a certain period such that the total utilization time of the sheet, i.e., the makespan of the schedule, is minimized. The 3D-STO involves a series of 2D rectangular packing problem (2D-RPP) for the whole scheduling period. If the processing time of each item is simply regarded as a space dimension for the packing, then the 3D-STO can be reduced to an NP-hard packing problem—3D strip packing problem (3D-SPP). The 3D-STO differs from the 3D-SPP in that the position and orientation of items can be changed over the processing time (the third dimension) such that the 3D-STO has a much larger search space compared with the 3D-SPP. Hereby, the optimal solution of the 3D-STO is better than or equal to the optimal solution of the 3D-SPP, and a possibly better solution could be found for the 3D-STO model. As a new model, there is no algorithm or benchmark in the literature. Moreover, the 2D-RPP and 3D-SPP algorithms are not suitable for solving the 3D-STO which considers the packing and scheduling simultaneously. Hereby, we propose a caving-degree-based scheduling algorithm (CDS) for the 3D-STO. This is the first proposed 3D-STO algorithm. We also formalize the problem as a mixed integer programming model and solve it by ILOG CPLEX. For evaluation, we provide a synthesized method to generate a total of 195 various benchmark instances with guillotine and nonguillotine cut constraints. The comparative results show that CDS is more effective than the CPLEX solver for the 3D-STO. Also, when comparing CDS for the 3D-STO and the adapted CDS for the 3D-SPP, we see that the new 3D-STO model can enhance the flexibility of the item arrangement and make maximum utilization of the sheet and time.