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We address the single container packing problem of a company that has to serve its customers by first placing the products in boxes and then loading the boxes into a container. We approach the problem by developing and solving mixed-integer linear models. Our models consider geometric constraints that feature non-overlapping constraints, box orientation constraints, dimensionality constraints, relative packing position constraints, and linearity constraints. We also develop an extension of the models by integrating load balance and the deviation of the center of gravity. We tested the models on a broad set of real instances involving up to 41 boxes and obtained optimal solutions in most cases and very small gaps when optimality could not be proven.
We present a general numerical solution method for control problems with PDE-defined state variables over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs (and if necessary a linearization scheme) to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level parameters:
First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.