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/ )

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Abstract
Irregular strip packing problems are present in a wide variety of industrial sectors, such as the garment, footwear, furniture and metal industry. The goal is to find a layout in which an object will be cut into small pieces with minimum raw-material waste. Once a layout is obtained, it is necessary to determine the path that the cutting tool has to follow to cut the pieces from the layout. In the latter, the goal is to minimize the cutting distance (or time). Although industries frequently use this solution sequence, the trade-off between the packing and the cutting path problems can significantly impact the production cost and productivity. A layout with minimum raw-material waste, obtained through the packing problem resolution, can imply a longer cutting path compared to another layout with more material waste but a shorter cutting path, obtained through an integrated strategy. Layouts with shorter cutting path are worthy of consideration because they may improve the cutting process productivity. In this paper, both problems are solved together using a biobjective matheuristic based on the Biased Random-Key Genetic Algorithm. Our approach uses this algorithm to select a subset of the no-fit polygons edges to feed the mathematical model, which will compute the layout waste and cutting path length. Solving both strip packing and cutting path problems simultaneously allows the decision-maker to analyze the compromise between the material waste and the cutting path distance. As expected, the computational results showed the trade-off’s relevance between these problems and presented a set of solutions for each instance solved. © 2023, Sociedade Brasileira de Pesquisa Operacional. All rights reserved.

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
Cutting and packing problems are hard combinatorial optimization problems that arise in several manufacturing and process industries or in their supply chains. The solution of these problems is not only a scientific challenge but also has a large economic impact, as it contributes to the reduction of one of the major cost factors for many production sectors, namely raw materials, together with a positive environmental impact. The explicit consideration of uncertainty when solving cutting and packing problems with optimization techniques is crucial for a wider adoption of research results by companies. However, current research has paid little attention to the role of uncertainty in these problems. In this paper, we review the existing literature on uncertainty in cutting and packing problems, propose a classification framework, and highlight the many research gaps and opportunities for scientific contributions.