A tailored two-phase constructive heuristic for the three-dimensional Multiple Bin Size Bin Packing Problem with transportation constraints

This paper considers the three-dimensional Multiple Bin Size Bin Packing Problem which consists in packing a set of cuboid boxes into containers of various shapes, while minimising unused space. The problem is extended to air cargo where the bins are Unit Load Devices, specially designed for fitting in aircraft. We developed a fast constructive heuristic able to manage the constraints to be met in transportation. The heuristic is split into two distinct phases. The first phase deals with the packing of boxes into identical bins using an extension of the Extreme Points which describe the possible interesting positions to accommodate boxes. During this phase, the fragility, stability and orientation of the boxes are taken into account as well as the special shape of the bins and their weight capacity. The second phase considers the multiple types of available bins. If necessary, the best loading pattern identified is enhanced with respect to weight distribution in post processing. After the description of the parametrisation, computational experiments are performed on data sets specially designed for this application. The heuristic requires only few seconds to achieve promising results in terms of filling rate

The Airline Container Loading Problem with Pickup and Delivery

This paper considers the loading optimization problem for a set of containers and pallets transported into a cargo aircraft that serves multiple airports. Because of pickup and delivery operations that occur at intermediate airports, this problem is simultaneously a Weight & Balance Problem and a Sequencing Problem. Our objective is to minimize fuel and handling operation costs. This problem is shown to be NP-hard. We resort to a mixed integer linear program. Based on real-world data from a professional partner (TNT Airways), we perform numerical experiments using a standard B&C library. This approach yields better solutions than traditional manual planning, which results in substantial cost savings.Optimization in air transpor

A relaxed approach to combinatorial problems in robustness and diagnostics

A range of procedures in both robustness and diagnostics require optimisation of a target functional over all subsamples of given size. Whereas such combinatorial problems are extremely difficult to solve exactly, something less than the global optimum can be ‘good enough’ for many practical purposes, as shown by example. Again, a relaxation strategy embeds these discrete, high-dimensional problems in continuous, low-dimensional ones. Overall, nonlinear optimisation methods can be exploited to provide a single, reasonably fast algorithm to handle a wide variety of problems of this kind, thereby providing a certain unity. Four running examples illustrate the approach. On the robustness side, algorithmic approximations to minimum covariance determinant (MCD) and least trimmed squares (LTS) estimation. And, on the diagnostic side, detection of multiple multivariate outliers and global diagnostic use of the likelihood displacement function. This last is developed here as a global complement to Cook’s (in J. R. Stat. Soc. 48:133–169, 1986) local analysis. Appropriate convergence of each branch of the algorithm is guaranteed for any target functional whose relaxed form is—in a natural generalisation of concavity, introduced here—‘gravitational’. Again, its descent strategy can downweight to zero contaminating cases in the starting position. A simulation study shows that, although not optimised for the LTS problem, our general algorithm holds its own with algorithms that are so optimised. An adapted algorithm relaxes the gravitational condition itself