Abstract
The container loading problem (CLP) is a combinatorial optimization problem for the spatial arrangement of cargo inside containers so as to maximize the usage of space. The algorithms for this problem are of limited practical applicability if real-world constraints are not considered, one of the most important of which is deemed to be stability. This paper addresses static stability, as opposed to dynamic stability, looking at the stability of the cargo during container loading. This paper proposes two algorithms. The first is a static stability algorithm based on static mechanical equilibrium conditions that can be used as a stability evaluation function embedded in CLP algorithms (e.g. constructive heuristics, metaheuristics). The second proposed algorithm is a physical packing sequence algorithm that, given a container loading arrangement, generates the actual sequence by which each box is placed inside the container, considering static stability and loading operation efficiency constraints.

Abstract
Two-dimensional rectangular strip packing problems belong to the broader class of Cutting and Packing (C&P) problems, in which small items are required to be cut from or packed on a larger object, so that the waste (unused regions of the large object) is minimized. C&P problems differ from other combinatorial optimization problems by the intrinsic geometric constraints: items may not overlap and have to be fully contained in the large object. This survey approaches the specific C&P problem in which all items are rectangles, therefore fully characterized by a width and a height, and the large object is a strip, i.e. a rectangle with a fixed width but an infinite height, being the problem’s goal to place all rectangles on the strip so that the height is minimized. These problems have been intensively and extensively tackled in the literature and this paper will focus on heuristic resolution methods. Both the seminal and the most recent approaches (from the last decade) will be reviewed, in a rather tutorial flavor, and classified according to their type: constructive heuristics, improvement heuristics with search over sequences and improvement heuristics with search over layouts. Building on this review, research gaps are identified and the most interesting research directions pointed out. © 2016 Brazilian Operations Research Society.

Abstract

Abstract
Two-dimensional irregular strip packing problems are cutting and packing problems where small pieces have to be cut from a larger object, involving a non-trivial handling of geometry. Increasingly sophisticated and complex heuristic approaches have been developed to address these problems but, despite the apparently good quality of the solutions, there is no guarantee of optimality. Therefore, mixed-integer linear programming (MIP) models started to be developed. However, these models are heavily limited by the complexity of the geometry handling algorithms needed for the piece non-overlapping constraints. This led to pieces simplifications to specialize the developed mathematical models. In this paper, to overcome these limitations, two robust MIP models are proposed. In the first model (DTM) the non-overlapping constraints are stated based on direct trigonometry, while in the second model (NFP - CM) pieces are first decomposed into convex parts and then the non-overlapping constraints are written based on nofit polygons of the convex parts. Both approaches are robust in terms of the type of geometries they can address, considering any kind of non-convex polygon with or without holes. They are also simpler to implement than previous models. This simplicity allowed to consider, for the first time, a variant of the models that deals with piece rotations. Computational experiments with benchmark instances show that NFP CM outperforms both DTM and the best exact model published in the literature. New real-world based instances with more complex geometries are proposed and used to verify the robustness of the new models.

Abstract
Solving nesting problems involves the waste minimisation in cutting processes, and therefore it is not only economically relevant for many industries but has also an important environmental impact, as the raw materials that are cut are usually a natural resource. However, very few exact approaches have been proposed in the literature for the nesting problem (also known as irregular packing problem), and the majority of the known approaches are heuristic algorithms, leading to suboptimal solutions. The few mathematical programming models known for this problem can be divided into discrete and continuous models, based on how the placement coordinates of the pieces to be cut are dealt with. In this paper, we propose an innovative semi-continuous mixed-integer programming model for two-dimensional cutting and packing problems with irregular shaped pieces. The model aims to exploit the advantages of the two previous classes of approaches and discretises the [GRAPHICS] -axis while keeping the [GRAPHICS] -coordinate continuous. The board can therefore be seen as a set of stripes. Computational results show that the model, when solved by a commercial solver, can deal with large problems and determine the optimal solution for smaller instances, but as it happens with discrete models, the optimal solution value depends on the discretisation step that is used.

Abstract
This paper describes the results of our collaboration with the leading Portuguese food retailer to address the shelf-space planning problem of allocating products to shop-floor shelves. Our challenge was to introduce analytical methods into the shelf-space planning process to improve the return on space and automate a process heavily dependent on the experience of the retailer's space managers. This led to the creation of GAP, a decision support system that the company's space-management team uses daily. We developed a modular operations research approach that systematically applies mathematical programming models and heuristics to determine the best layout of products on the shelves. GAP combines its analytical strength with an ability to incorporate different types of merchandising rules to balance the tradeoff between optimization and customization.