The nesting problem, also known as irregular packing problem, belongs to the generic class of cutting and packing (C&P) problems. It di↵ers from other 2-D C&P problems in the irregular shape of the pieces. This paper proposes a new mixed-integer model in which binary decision variables are associated with each discrete point of the board (a dot) and with each piece type. It is much more flexible than previously proposed formulations and solves to optimality larger instances of the nesting problem, at the cost of having its precision dependent on board discretization. To date no results have been published concerning optimal solutions for nesting problems with more than 7 pieces. We ran computational experiments on 45 problem instances with the new model, solving to optimality 34 instances with a total number of pieces ranging from 16 to 56, depending on the number of piece types, grid resolution and the size of the board. A strong advantage of the model is its insensitivity to piece and board geometry, making it easy to extend to more complex problems such as non-convex boards, possibly with defects. Additionally, the number of binary variables does not depend on the total number of pieces but on the number of piece types, making the model particularly suitable for problems with few piece types. The discrete nature of the model requires a trade-o↵ between grid resolution and problem size, as the number of binary variables grows with the square of the selected grid resolution and with board size.

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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