On the $$L$$ L -approach for generating unconstrained two-dimensional non-guillotine cutting patterns

Higher order block heuristics for 2D pallet loading problems with multiple box inputs

Greedy algorithms for a class of knapsack problems with binary weights

Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects are axis-parallel rectangles. However, even for the case of axis-parallel squares this question is still open. In this paper, we answer the latter affirmatively. Our result is constructive and holds even in a more general setting where the squares have weights and the goal is to save as much weight as possible. We further show that when solving the more general question for rectangles affirmatively with only axis-parallel cuts, this would yield a combinatorial O(1)-approximation algorithm for the Maximum Independent Set of Rectangles problem, and would thus solve a long-standing open problem. In practical applications, like the mentioned carpentry and many other settings, we can usually place the items freely that we want to cut out, which gives rise to the two-dimensional guillotine knapsack problem: Given a collection of axis-parallel rectangles without presumed coordinates, our goal is to place as many of them as possible in a square-shaped knapsack respecting the constraint that the placed objects can be separated by a sequence of guillotine cuts. Our main result for this problem is a quasi-PTAS, assuming the input data to be quasi-polynomially bounded integers. This factor matches the best known (quasi-polynomial time) result for (non-guillotine) two-dimensional knapsack.

Enhancing pallet load stability: A MILP model for the Manufacturer’s Pallet Loading Problem with interlocking constraints

The two-dimensional knapsack problem with splittable items in stacks

This study presents a two-phase algorithm for solving the Manufacturer's Pallet Loading Problem (MPLP) while considering overhang and stability constraints using a block approach. The MPLP involves determining a loading pattern that can load the most identical rectangular boxes onto a larger rectangular pallet. The proposed model is developed to obtain the maximum number of boxes that could be loaded onto a layer considering and without considering the overhang according to eight predetermined block arrangements (BAs) in the first phase. Then, the obtained overhang solutions are checked against the stability constraint in the second phase. Finally, the maximum number of boxes per layer is determined based on the results from the two phases. The validity and the performance of the proposed algorithm have been tested on available datasets in the literature. The proposed algorithm increased the number of boxes per layer; one to six boxes while considering overhang and stability, improving the pallet area utilization by 2.5% up to 14.3% for the tested datasets.

The paper discusses the complexities of the air cargo loading problem and provides an effective approach based on the proposed heuristic to tackle the problem. The primitive class of problems like general bin packing, cutting stock and knapsack problems can be taken as the basic step towards the air-cargo loading problem. The discussed problem is very different in its nature in respect to the general pallet, container or truck loading problem. Generally, the container loading problems consider only the rectangular type of containers to load the items whereas in case of air-cargos, the shape of containers are very different such as irregular pentagon, hexagon etc. Apart from this, the objective of air cargo loading problem discussed in this paper is based on the minimization of the shipping cost of the container keeping in mind the number of constraints involved in loading. In past research literatures, most of the container loading problems are aimed to pack the container in such a way so that either the number of containers is minimized or the volume of the packed items are maximized. In this paper the objective function includes the shipment cost as the primary goal and the volume and weight of the containers are taken as constraints. A very interesting finding can be noticed in this paper which states that even if the container's volume is properly utilized the cost of shipment may be increased.

A universal approach for continuous or discrete nonlinear programmings with multiple variables and constraints

This paper presents the results of an empirical study of the effects of coefficient correlation structure and constraint slackness settings on the performance of solution procedures on synthetic two-dimensional knapsack problems (2KP). The population correlation structure among 2KP coefficients, the level of constraint slackness, and the type of correlation (product moment or rank) are varied in this study. Representative branch-and-bound and heuristic solution procedures are used to investigate the influence of these problem parameters on solution procedure performance. Population correlation structure, and in particular the interconstraint component of the correlation structure, is found to be a significant factor influencing the performance of both the algorithm and the heuristic. In addition, the interaction between constraint slackness and population correlation structure is found to influence solution procedure performance.

This paper introduces the three‐dimensional case picking problem (3D‐CPP) and proposes a multistart biased‐randomized algorithm (BRA) to solve it. The 3D‐CPP combines two important topics in modern warehouse logistics: the pallet loading problem and the routing of pickers in manual warehouses. The proposed optimization procedure aims at minimizing the overall distance traveled by the pickers, and is achieved by combining a routing problem (i.e., the order in which picking positions are visited) with a loading problem (i.e., the way in which cases are placed onto the pallet). We also consider additional constraints regarding the weight, vertical support, and strength of the cases. In order to solve this problem, we first propose a constructive heuristic which combines routing and packing procedures. This initial heuristic is then extended into a multistart BRA by employing a skewed probability distribution to introduce a certain degree of randomness during the solution‐construction process. A series of computational experiments allow us to assess the quality of the proposed approach, through a comparison with other algorithms as well as using real‐life data provided by an industrial partner.

Neste artigo apresentamos métodos exatos, baseados em relaxações Lagrangiana e surrogate, com bom desempenho para resolver o problema de carregamento de paletes do produtor. Este problema consiste em arranjar ortogonalmente e sem sobreposição o máximo número de retângulos de dimensões (l,w) ou (l,w) sobre um retângulo maior (L,W). Os métodos propostos são procedimentos de busca em árvore do tipo branch and bound que, em cada nó, utilizam limitantes derivados de relaxações Lagrangiana e/ou surrogate de uma formulação de programação linear 0-1. Algoritmos de otimização do subgradiente são usados para otimizar estes limitantes. São aplicados ainda testes de redução do problema e heurísticas Lagrangiana e surrogate para se obter boas soluções factíveis na otimização do subgradiente. Testes computacionais foram realizados utilizando exemplos da literatura e exemplos reais de uma transportadora. Os resultados mostram que um dos métodos propostos é competitivo com outros métodos da literatura, incluindo o software GAMS/CPLEX.

This paper concerns the use of strategic oscillation (SO) for packing problems. The SO algorithm spends a certain time exploring infeasible solutions rather than restricting the search to feasible solutions only. The main features of the algorithm are explained and suggestions on different implementations are given. The pallet-loading problem is probably the simplest form of packing problem and is thus suitable for initial experiments with SO. The results obtained suggest that SO is successful for this problem and that it will probably be fruitful to extend the use of SO to other, harder, packing problems.

A two-step evolutionary algorithm for the distributor’s pallet loading problem with multi-size pallets

This article addresses several variants of the two-dimensional bin packing problem. In the most basic version of the problem it is intended to pack a given number of rectangular items with given sizes in rectangular bins in such a way that the number of bins used is minimized. Different heuristic approaches (greedy, local search, and variable neighbourhood descent) are proposed for solving four guillotine two-dimensional bin packing problems. The heuristics are based on the definition of a packing sequence for items and in a set of criteria for packing one item in a current partial solution. Several extensions are introduced to deal with issues pointed out by two furniture companies. Extensive computational results on instances from the literature and from the two furniture companies are reported and compared with optimal solutions, solutions from other five (meta)heuristics and, for a small set of instances, with the ones used in the companies.

We consider a two-dimensional problem in which one is required to split a given rectangular bin into the smallest number of items. The resulting items must be squares to be packed, without overlapping, into the bin so as to cover all the given rectangle. We present a mathematical model and a heuristic algorithm that is proved to find the optimal solution in some special cases. Then, we introduce a relaxation of the problem and present different exact approaches based on this relaxation. Finally, we report computational experiments on the performances of the algorithms on a large set of randomly generated instances.