Abstract
In this paper, we consider the two-dimensional Variable-Sized Cutting Stock Problem (2D-VSCSP) with guillotine constraint, applied to the home textile industry. This is a challenging class of real-world prob-lems where, given a set of predefined widths of fabric rolls and a set of piece types, the goal is to de-cide the widths and lengths of the fabric rolls to be produced, and to generate the cutting patterns to cut all demanded pieces. Each piece type considered has a rectangular shape with a specific width and length and a fixed demand to be respected. The main objective function is to minimize the total amount of the textile materials produced/cut to satisfy the demand. According to Wascher, Hau ss ner, & Schu-mann (2007), the addressed problem is a Cutting Stock Problem (CSP), as the demand for each item is greater than one. However, in the real-world application at stake, the demand for each item type is not very high (below ten for all item types). Therefore, addressing the problem as a Bin-Packing Problem (BPP), in which all items are considered to be different and have a unitary demand, was a possibility. For this reason, two approaches to solve the problems were devised, implemented, and tested: (1) a CSP model, based on the well-known Lodi and Monaci (2003) model (3 variants), and (2) an original BPP-based model. Our research shows that, for this level of demand, the new BPP model is more competitive than CSP models. We analyzed these different models and described their characteristics, namely the size and the quality of the linear programming relaxation bound for solving the basic mono-objective variant of the problem. We also propose an epsilon-constraint approach to deal with a bi-objective extension of the problem, in which the number of cutting patterns used must also be minimized. The quality of the models was evaluated through computational experiments on randomly generated instances, yielding promising results.(c) 2022 Published by Elsevier B.V.

Abstract
Cutting and packing problems are challenging combinatorial optimization problems that have many rel-evant industrial applications and arise whenever a raw material has to be cut into smaller parts while minimizing waste, or products have to be packed, minimizing the empty space. Thus, the optimal solution to these problems has a positive economic and environmental impact. In many practical applications, both the raw material and the cut parts have a rectangular shape, and cut-ting plans are generated for one raw material rectangle (also known as plate) at a time. This is known in the literature as the (two-dimensional) rectangular cutting problem. Many variants of this problem may arise, led by cutting technology constraints, raw-material characteristics, and different planning goals, the most relevant of which are the guillotine cuts. The absence of the guillotine cuts imposition makes the problem harder to solve to optimality.Based on the Floating-Cuts paradigm, a general and flexible mixed-integer programming model for the general rectangular cutting problem is proposed. To the best of our knowledge, it is the first mixed inte-ger linear programming model in the literature for both non-guillotine and guillotine problems. The basic idea of this model is a tree search where branching occurs by successive first-order non-guillotine-type cuts. The exact position of the cuts is not fixed, but instead remains floating until a concrete small rect-angle (also known as item) is assigned to a child node. This model does not include decision variables either for the position coordinates of the items or for the coordinates of the cuts. Under this framework, it was possible to address various different variants of the problem.Extensive computational experiments were run to evaluate the model's performance considering 16 dif-ferent problem variants, and to compare it with the state-of-the-art formulations of each variant. The results confirm the power of this flexible model, as, for some variants, it outperforms the state-of-the-art approaches and, for the other variants, it presents results fairly close to the best approaches. But, even more importantly, this is a new way of looking at these problems which may trigger even better approaches, with the consequent economic and environmental benefits.

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.

Abstract
Cutting and packing problems have been widely studied in the last decades, mainly due to the variety of industrial applications where the problems emerge. This paper presents an overview of the solution approaches that have been proposed for solving two-dimensional rectangular cutting and packing problems. The main emphasis of this work is on two distinct problems that belong to the cutting and packing problem family. The first problem aims to place onto an object the maximum-profit subset of items, that is, output maximization, while the second one aims to place all the items using as few identical objects as possible, that is, input minimization. The objective of this paper is not to be exhaustive but to provide a solid grasp on two-dimensional rectangular cutting and packing problems by describing their most important solution approaches.

Abstract
Interest in integrating lot-sizing and cutting stock problems has been increasing over the years. This integrated problem has been applied in many industries, such as paper, textile and furniture. Yet, there are only a few studies that acknowledge the importance of uncertainty to optimise these integrated decisions. This work aims to address this gap by incorporating demand uncertainty through stochastic programming and robust optimisation approaches. Both robust and stochastic models were specifically conceived to be solved by a column generation method. In addition, both models are embedded in a rolling-horizon procedure in order to incorporate dynamic reaction to demand realisation and adapt the models to a multistage stochastic setting. Computational experiments are proposed to test the efficiency of the column generation method and include a Monte Carlo simulation to assess both stochastic programming and robust optimisation for the integrated problem. Results suggest that acknowledging uncertainty can cut costs by up to 39.7%, while maintaining or reducing variability at the same time.