Abstract
We address two two-dimensional bin packing problems where the bins are rectangular and have the same size. The items are also rectangular and all of them must be packed with the objective of minimizing the number of bins. In the first problem, the two-stage problem, the items must be packed in levels. In the second problem, the restricted 3-stage problem, items can be grouped in stacks which are packed in levels. We propose a new decomposition model where subproblems are associated with the item that initializes a bin. The decomposition is solved by a heuristic which combines (perturbed) column generation, local search, beam branch-and-price, and the use of a general purpose mixed integer programming solver. This approach is closely related with SearchCol, a framework for solving integer programming / combinatorial optimization decomposition models. Computational results with 500 instances from the literature show that the proposed hybrid heuristic is efficient in obtaining high quality solutions. It uses more 8 and 17 bins than the 7364 and 7340 bins of a compact model from the literature for the 2 and 3-stage problems, respectively, while the sum of the time spent for all instances is 35% and 58% of the time spent by the compact model.

Abstract

Abstract
In road freight transport, a loaded vehicle with a distribution route and a compliant load balance at the depot can become non-compliant during the route, since the total weight of the cargo and its centre of gravity change with each delivery. Nowadays, vehicles circulating on our roads either undermine safety regulations or lack operational efficiency when these regulations are taken into account and cargo is extensively rearranged after each delivery. This issue has been completely ignored both in the vehicle routing literature and in the container loading literature. The aim of this work is to provide tools capable of ensuring that a cargo arrangement is load balanced along the complete distribution trip. It proposes a multi-drop load balance recovery algorithm (MDLBRA), which seeks to ensure that, when both a complete route and the respective cargo arrangement are provided, the boxes to be removed from the cargo arrangement at the depot and the boxes to be rearranged at each customer are identified, allowing the cargo to remain balanced after every delivery. It is important to notice that a MDLBRA is not a container loading algorithm: a MDLBRA modifies solutions generated by any container loading algorithm so that load balance is guaranteed when the truck leaves the depot and during the entire distribution route. A mixed integer linear programming (MILP) model is proposed to balance the cargo at each customer stop. The MILP model incorporates load distribution diagram constraints in order to determine the feasible domain for the location of the centre of gravity of the cargo arrangement, taking into account the regulatory requirements and the technical characteristics of the vehicle. Extensive computational experiments show that a MDLBRA can be used in practical contexts, as the MILP model was able to find a solution in less than ten minutes in 93% of the unbalanced test instances.

Abstract
In this paper, we explore the use of reference values (predictors) for the optimal objective function value of hard combinatorial optimization problems, instead of bounds, obtained by data mining techniques, and that may be used to assess the quality of heuristic solutions for the problem. With this purpose, we resort to the rectangular two-dimensional strip-packing problem (2D-SPP), which can be found in many industrial contexts. Mostly this problem is solved by heuristic methods, which provide good solutions. However, heuristic approaches do not guarantee optimality, and lower bounds are generally used to give information on the solution quality, in particular, the area lower bound. But this bound has a severe accuracy problem. Therefore, we propose a data mining-based framework capable of assessing the quality of heuristic solutions for the 2D-SPP. A regression model was fitted by comparing the strip height solutions obtained with the bottom-left-fill heuristic and 19 predictors provided by problem characteristics. Random forest was selected as the data mining technique with the best level of generalisation for the problem, and 30,000 problem instances were generated to represent different 2D-SPP variations found in real-world applications. Height predictions for new problem instances can be found in the regression model fitted. In the computational experimentation, we demonstrate that the data mining-based framework proposed is consistent, opening the doors for its application to finding predictions for other combinatorial optimisation problems, in particular, other cutting and packing problems. However, how to use a reference value instead of a bound, has still a large room for discussion and innovative ideas. Some directions for the use of reference values as a stopping criterion in search algorithms are also provided.

Abstract
In this paper, we address the Constrained Three-dimensional Guillotine Cutting Problem (C3GCP), which consists of cutting a larger cuboid block (object) to produce a limited number of smaller cuboid pieces (items) using orthogonal guillotine cuts only. This way, all cuts must be parallel to the object's walls and generate two cuboid sub-blocks, and there is a maximum number of copies that can be manufactured for each item type. The C3GCP arises in industrial manufacturing settings, such as the cutting of steel and foam for mattresses. To model this problem, we propose a new compact mixed-integer non-linear programming (MINLP) formulation by extending its two-dimensional version, and develop a mixed-integer linear programming (MILP) version. We also propose a new model for a particular case of the problem which considers 3-staged patterns. As a solution method, we extend the algorithm of Wang (1983) to the three-dimensional case. We emphasise that the C3GCP is different from 3D packing problems, namely from the Container Loading Problem, because of the guillotine cut constraints. All proposed approaches are evaluated through computational experiments using benchmark instances. The results show that the approaches are effective on different types of instances, mainly when the maximum number of copies per item type is small, a situation typically encountered in practical settings with low demand for each item type. These approaches can be easily embedded into existing expert systems for supporting the decision-making process.

Abstract
The backroom of retail stores has structural differences when compared with other warehouses and distribution centres, which are more traditionally studied in the literature. This paper presents a mathematical optimisation approach for an unequal area facility layout problem, applied in designing the backroom layout in grocery retail. A set of rectangular facilities (backroom departments) with given area requirements has to be placed, without overlapping, on a limited floor space (backroom area), which can have a regular or an irregular shape. The objective is to find the location and format of the storage departments, such that the walking distances in the store by store employees are minimised. The proposed approach is tested in a European grocery retailer. In the computational experiments, several real store layouts are compared with the ones suggested by the proposed model. The decrease in the walking distances is, on average, 30 percent. In order to understand what the current designers' strategy is, a set of scenarios was created and compared with the real layouts. Each scenario ignores a characteristic of the problem. The goal is to understand what aspect designers are currently discarding. The findings indicate that, currently, designers neglect the different replenishment frequencies of storage departments.