In this paper, we present an integer linear programming model for the vehicle routing problem that considers real-world three-dimensional (3D) loading constraints. In this problem, a set of customers make requests of goods that are wrapped up in boxes, and the objective is to find minimum cost delivery routes for a set of identical vehicles that, departing from a depot, visit all customers only once and return to the depot. Apart from the usual 3D container loading constraints that ensure the boxes are packed completely inside the vehicles and the boxes do not overlap each other in each vehicle, the problem also takes into account constraints related to the vertical stability of the cargo, multidrop situations, and load-bearing strength of the boxes (including fragility). Computational tests with the proposed model were performed using an optimization solver embedded into a modeling language. The results validate the model and show that it is only able to handle problems of a moderate size. However, this model will be useful to motivate other researchers to explore approximate solution approaches to solve this problem, such as decomposition methods, relaxation methods, heuristics, among others, as well as to treat other variants of the problem, such as when time windows or a heterogeneous fleet are present, among others.

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In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an L-approach for packing rectangles into larger rectangles and L-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50 000 instances). It is also effective for solving the instances of problem set Cover III (almost 100 000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes.

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This paper introduces an evolutionary algorithm as a procedure to solve the Synchronized and Integrated Two-Level Lot Sizing and Scheduling Problem (SITLSP). This problem can be found in some industrial settings, mainly soft drink companies, where the production process involves two interdependent levels with decisions concerning raw material storage and soft drink bottling. The challenge is to simultaneously determine the lot-sizing and scheduling of raw materials in tanks and soft drinks in bottling lines, where setup costs and times depend on the previous items stored and bottled. A multi-population genetic algorithm approach with a novel representation of solutions for individuals and a hierarchical ternary tree structure for populations is proposed. Computational tests include comparisons with an exact approach for small-tomoderate sized instances and with real-world production plans provided by a manufacturer.