Experiments with a strategic oscillation algorithm for the pallet loading problem

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

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

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.

Packing problems deal with loading of a set of items (objects) into a set of boxes (containers) in order to optimize a performance criterion under various constraints. With the advance of RFID technologies and investments in IT infrastructures companies now have access to the necessary data that can be utilized in cost reduction of packing processes. Therefore bin packing and container loading problems are becoming more popular in recent years. In this research we propose a beam search algorithm to solve a packing problem that we encountered in a real world project. The 3D-MBSBPP (Multiple Bin Sized Bin Packing Problem) that we present and solve has not been analyzed in literature before, to the best of our knowledge. We present the performance of our proposed beam search algorithm in terms of both cost and computational time in comparison to a greedy algorithm and a tree search enumeration algorithm.

A reinforcement learning guided hybrid evolutionary algorithm for the latency location routing problem

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.

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.

Many cutting problems on two- or three-dimensional objects require that the cuts be orthogonal and of guillotine type. However, there are applications in which the cuts must be orthogonal but need not be of guillotine type. In this paper we focus on the latter type of cuts on rectangular bins. We investigate the so-called \(L\)-approach, introduced by Lins et al. (J Oper Res Soc 54:777–789, 2003), which has been used to tackle, among others, the pallet loading problem and the two-dimensional unconstrained knapsack problem. This approach concerns a method to generate two \(L\)-shaped pieces from an \(L\)-shaped piece. More recently, Birgin et al. (J Oper Res Soc 63(2):183–200, 2012), raised two questions concerning this approach. The first question is whether one may restrict only to cuts on raster points (rather than on all discretization points), without loss in the quality of the solution. We prove that the answer to this question is positive. The second question is whether the \(L\)-approach is an optimal method to solve the unconstrained knapsack problem. We show an instance of the problem for which this approach fails to find an optimum solution.

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

Geometric Covering and Packing problems have been extensively studied in the last few decades and have applications in diverse areas. Several variants and generalizations of these problems have been studied recently. In this paper, we look at the following covering variants where we require that each point is “uniquely” covered, i.e., it is covered by exactly one object: Unique Coverage problem, where we want to maximize the number of uniquely covered points and Exact Cover problem, where we want to uniquely cover every point and minimize the number of objects used for covering. We also look at the following generalizations: Multi Cover problem, a generalization of Set Cover, the objective is to select the minimum subset of objects with the constraint that each input point p is covered by at least $$d_p$$ objects in the solution, where $$d_p$$ is the demand of point p. And Shallow Packing problem, a generalization of Packing problem, where we want to select the maximum subset of objects with the constraint that any point in the plane is contained in at most k objects in the solution. The above problems are NP-hard even for unit squares in the plane. Thus, the focus has been on obtaining good approximation algorithms. Local search has been quite successful in the recent past in obtaining good approximation algorithms for a wide variety of problems. We consider the Unique Coverage and Multi Cover problems on non-piercing objects, which is a broad class that includes squares, disks, pseudo-disks, etc. and show that the local search algorithm yields a PTAS approximation under the assumption that the depth of every input point is at most some fixed constant. For Unique Coverage we further assume that every object has at most a constant degree. For the Shallow Packing problem, we show that the local search algorithm yields a PTAS approximation for objects with sub-quadratic union complexity, which is a very broad class of objects that even includes non-piercing objects. For the Exact Cover problem, we show that finding a feasible solution is NP-hard even for unit squares in the plane, thus negating the existence of polynomial time approximation algorithms.

This paper proposes an efficient algorithm, with a reduced number of parameters, for solving the two‐dimensional loading‐capacitated vehicle routing problem (2L‐CVRP). This problem combines two of the most important issues in logistics, that is, vehicle routing and packing problems. Our approach contemplates unrestricted loading including the possibility of applying 90° rotations to each rectangular‐shaped item while loading it into the vehicle, which is a realistic assumption seldom considered in the existing literature. The algorithm uses a multistart approach that is designed to avoid local minima and also to make the algorithm an easily parallelizable one. At each restart, a biased randomization of a savings‐based routing algorithm is combined with an enhanced version of a classical packing heuristic to produce feasible good solutions for the 2L‐CVRP. The proposed algorithm has been compared with the classical benchmarks for two different 2L‐CVRP variants, that is, with and without item rotations. Experimental results show that our approach outperforms several best‐known solutions from previous work, both in terms of quality and the computational time needed to obtain them.

New relaxations are developed in this paper for problems of optimal packing of small (rectangular‐shaped) pieces within one or several larger containers. Based on these relaxations tighter bounds for the Container Loading Problem (CLP) and the Multi‐Container Loading Problem (MCLP) are obtained.The new relaxations for the CLP and MCLP lead to linear programming problems. A corresponding solution approach is discussed which is based on a column generation technique. Results of computational tests are also given.

Using a Bin Packing Approach for Stowing Hazardous Containers into Containerships.- Dynamic Packing with Side Constraints for Datacenter Resource Management.- Packing Optimization of Free-Form Objects in Engineering Design.- A Modeling-Based Approach for Non-Standard Packing Problems.- CAST: A Successful Project in Support of the International Space Station Logistics.- Cutting and Packing Problems with Placement Constraints.- A Container Loading Problem MILP-based Heuristics Solved by CPLEX:An Experimental Analysis.- Automatic Design of Optimal LED Street Lights.- Approximate Packing: Integer Programming Models, Valid Inequalities and Nesting.- Exploiting Packing Components in General-Purpose Integer Programming Solvers.- Robust Designs for Circle Coverings of a Square.- Batching-based Approaches for Optimized Packing of Jobs in the Spatial Scheduling Problem.- Optimized Object Packings Using Quasi-Phi-Functions.- Graph Coloring Models and Metaheuristics for Packing Applications.

This research effort presents a tabu search algorithm to solve the dynamic airlift loading problem. Given a set of palletized cargo items which require transportation from an aerial port of embarkation to an aerial port of debarkation within a pre-specified time frame, the dynamic airlift loading problem seeks to partition the pallets into aircraft loads, select an efficient and effective subset of aircraft from available aircraft, and assign the pallets to allowable positions on those aircraft. The dynamic airlift loading problem differs from many partitioning and packing problems described in the literature because, in addition to spatial constraints, factors such as allowable cabin load, balance restrictions, and temporal restrictions on cargo and aircraft are included. The algorithm developed in this research, the dynamic airlift loading problem-tabu search, was tested on a variety of problem instances. Since real-world solutions are hand generated by subject matter experts and no previous research effort has solved this specific problem, the algorithmic results are compared to compute lower bounds on the number of aircraft trips required.