Abstract
As in many other combinatorial optimisation problems, research on nesting problems (aka irregular packing problems) has evolved around the dichotomy between continuous (time consuming) and discrete (memory consuming) representations of the solution space. Recent research has been devoting increasing attention to discrete representations for the geometric layer of nesting problems, namely in mathematical programming-based approaches. These approaches employ conventional regular meshes, and an increase in their precision has a high computational cost. In this paper, we propose a data structure to represent non-regular meshes, based on the geometry of each piece. It supports non-regular discrete geometric representations of the shapes, and by means of the proposed data structure, the discretisation can be easily adapted to the instances, thus overcoming the precision loss associated with discrete representations and consequently allowing for a more efficient implementation of search methods for the nesting problem. Experiments are conducted with the dotted-board model - a recently published mesh-based binary programming model for nesting problems. In the light of both the scale of the instances, which are now solvable, and the quality of the solutions obtained, the results are very promising.

Abstract
In cutting processes, one of the strategies to reduce raw material waste is to generate leftovers that are large enough to return to stock for future use. The length of these leftovers is important since waste is expected to be minimal when cutting these objects in the future. However, in several situations, future demand is unknown and evaluating the best length for the leftovers is challenging. Furthermore, it may not be economically feasible to manage a stock of leftovers with multiple lengths that may not result in minimal waste when cut. In this paper, we approached the cutting stock problem with the possibility of generating leftovers as a two-stage stochastic program with recourse. We approximated the demand levels for the different items by employing a finite set of scenarios. Also, we modeled different decisions made before and after uncertainties were revealed. We proposed a mathematical model to represent this problem and developed a column generation approach to solve it. We ran computational experi-ments with randomly generated instances, considering a representative set of scenarios with a varying probability distribution. The results validated the efficiency of the proposed approach and allowed us to derive insights on the value of modeling and tackling uncertainty in this problem. Overall, the results showed that the cutting stock problem with usable leftovers benefits from a modeling approach based on sequential decision-making points and from explicitly considering uncertainty in the model and the solution method. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Abstract
When dealing with cutting problems, the generation of usable leftovers proved to be a good strategy for decreasing material waste. Focusing on practical applications, the main challenge in the implementation of this strategy is planning the cutting process to produce leftovers with a high probability of future use without complete information about the demand for any ordered items. We addressed the two-dimensional cutting stock with usable leftovers and uncertainty in demand, a complex and relevant problem recurring in companies due to the unpredictable occurrence of customer orders. To deal with this problem, a two-stage formulation that approximates the uncertain demand by a finite set of possible scenarios was proposed. Also, we proposed a matheuristic to support decision-makers by providing good-quality solutions in reduced time. The results obtained from the computational experiments using instances from the literature allowed us to verify the matheuristic performance, demonstrating that it can be an efficient tool if applied to real-life situations.

Abstract
This paper studies the process of cutting steel bars in a truck suspension factory with the objective of reducing its inventory costs and material losses. A mathematical model is presented that focuses on decisions for a medium-term horizon (4 periods of 2 months). This approach addresses the one-dimensional 3-level integrated lot sizing and cutting stock problem, considering demand, inventory costs and stock level limits for bars (objects-level 1), springs (items-level 2) and spring bundles (final products-level 3), as well as the acquisition of bars as a decision variable. The solution to the proposed mathematical model is reached through an optimization package, using column generation along with a method for achieving integer solutions. The results obtained with real data demonstrate that the method provides significantly better solutions than those carried out at the company, whilst using reduced computational time. Additionally, the application of tests with random data enabled the analysis of both the effect of varying parameters in the solution, which provides managerial insights, and the overall performance of the method.

Abstract
The irregular strip packing problem consists of cutting a set of convex and non-convex two-dimensional polygonal pieces from a board with a fixed height and infinite length. Owing to the importance of this problem, a large number of mathematical models and solution methods have been proposed. However, only few papers consider that the pieces can be rotated at any angle in order to reduce the board length used. Furthermore, the solution methods proposed in the literature are mostly heuristic. This paper proposes a novel mixed integer quadratically-constrained programming model for the irregular strip packing problem considering continuous rotations for the pieces. In the model, the pieces are allocated on the board using a reference point and its allocation is given by the translation and rotation of the pieces. To reduce the number of symmetric solutions for the model, sets of symmetry-breaking constraints are proposed. Computational experiments were performed on the model with and without symmetry-breaking constraints, showing that symmetry elimination improves the quality of solutions found by the solution methods. Tests were performed with instances from the literature. For two instances, it was possible to compare the solutions with a previous model from the literature and show that the proposed model is able to obtain numerically accurate solutions in competitive computational times.

Abstract
Brazil is the largest sugarcane producer in the world and the leader in the production of sugar and ethanol. Although sugarcane has become an important factor in the Brazilian economy, cultivation has presented many issues, for example, the problems due to burning before the manual harvest. The Brazilian authorities have approved a law that prohibits this practice and mechanized harvesting has thus become the most fitting approach. Given this development, areas for sugarcane plantation must be properly rebuilt to accommodate the new way of harvesting. The main characteristic demanded of sugarcane plots to use harvesting machines is that they must be rectangular. In the present paper, we propose a methodology for dividing the plantation area into plots and planning their allocation so as to accommodate mechanized harvesting. In view of the requirement for plots to be rectangular, we represented this problem as a two-dimensional cutting problem, and to find a solution we adopted the AND/OR graph approach. The computational experiments were conducted using real cases, and the proposed strategy was shown to perform well.