Solving the Multi-objective 2-Dimensional Vector Packing Problem Using ϵ-constraint Method

In this paper, an exact method is designed to solve the multi-objective 2-dimensional vector packing problem. The algorithm is an adapted version of an efficient ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon$$\end{document}-constraint method which proves its efficiency in solving a large variety of multi-objective optimization problems. This method is based on a clever decomposition of the initial problem into sub-problems which are iteratively solved through mathematical programming. To accelerate the search process, we propose a new integer programming model for solving the multi-objective 2-dimensional vector packing problem based on the compact model for the bin packing problem with fragile objects. Instead of scanning all possible solutions, we consider the solutions while solving a Subset-Sum Problem. Hence, non-useful subproblems are avoided and thus the search space is reduced. An experimental study is performed based instances from the literature. A comparison between the exact method and a grounded metaheuristic which provides good results in solving the multi-objective 2-dimensional vector packing problem

Robotic Learning the Sequence of Packing Irregular Objects from Human Demonstrations

We address the unsolved task of robotic bin packing with irregular objects, such as groceries, where the underlying constraints on object placement and manipulation, and the diverse objects' physical properties make preprogrammed strategies unfeasible. Our approach is to learn directly from expert demonstrations in order to extract implicit task knowledge and strategies to achieve an efficient space usage, safe object positioning and to generate human-like behaviors that enhance human-robot trust. We collect and make available a novel and diverse dataset, BoxED, of box packing demonstrations by humans in virtual reality. In total, 263 boxes were packed with supermarket-like objects by 43 participants, yielding 4644 object manipulations. We use the BoxED dataset to learn a Markov chain to predict the object packing sequence for a given set of objects and compare it with human performance. Our experimental results show that the model surpasses human performance by generating sequence predictions that humans classify as human-like more frequently than human-generated sequences.Comment: 8 pages, 7 figure

Container Loading Problems: A State-of-the-Art Review

Container loading is a pivotal function for operating supply chains efficiently. Underperformance results in unnecessary costs (e.g. cost of additional containers to be shipped) and in an unsatisfactory customer service (e.g. violation of deadlines agreed to or set by clients). Thus, it is not surprising that container loading problems have been dealt with frequently in the operations research literature. It has been claimed though that the proposed approaches are of limited practical value since they do not pay enough attention to constraints encountered in practice.In this paper, a review of the state-of-the-art in the field of container loading will be given. We will identify factors which - from a practical point of view - need to be considered when dealing with container loading problems and we will analyze whether and how these factors are represented in methods for the solution of such problems. Modeling approaches, as well as exact and heuristic algorithms will be reviewed. This will allow for assessing the practical relevance of the research which has been carried out in the field. We will also mention several issues which have not been dealt with satisfactorily so far and give an outlook on future research opportunities

Depth-Optimized Reversible Circuit Synthesis

In this paper, simultaneous reduction of circuit depth and synthesis cost of reversible circuits in quantum technologies with limited interaction is addressed. We developed a cycle-based synthesis algorithm which uses negative controls and limited distance between gate lines. To improve circuit depth, a new parallel structure is introduced in which before synthesis a set of disjoint cycles are extracted from the input specification and distributed into some subsets. The cycles of each subset are synthesized independently on different sets of ancillae. Accordingly, each disjoint set can be synthesized by different synthesis methods. Our analysis shows that the best worst-case synthesis cost of reversible circuits in the linear nearest neighbor architecture is improved by the proposed approach. Our experimental results reveal the effectiveness of the proposed approach to reduce cost and circuit depth for several benchmarks.Comment: 13 pages, 6 figures, 5 tables; Quantum Information Processing (QINP) journal, 201

Method for loading cargo trucks using two-dimensional packing algorithms

The paper describes the method for optimization of loading cargo trucks using two-dimensional packing algorithms. The point of this method is to reduce loading cargo problem to two-dimensional packing problem. This problem can be solved by using of various algorithms. There is analysis of several algorithms that are most often used in practical calculations of objects distribution in 2D space in this paper. The object of this study is transport of the metal processing company and its products (cargo). PHP programming language, MySQL database, and Apache web server are used to create client application. The interface developed using HTML5, CSS and javascript. © 2019 IOP Publishing Ltd. All rights reserved.The work was supported by Act 211 Government of the Russian Federation, contract № 02.A03.21.0006

Métodos de análise da complexidade no problema de empacotamento de paletes do distribuidor

In the pallet loading problem, one of the main goals is to allocate the highest number of boxes as possible, to minimize empty spaces in the pallet. Those empty spaces are called trim-loss. If all boxes have a rectangular shape, which is the most common one, it is possible to pack them so that their faces are coincident with themselves. By doing that, the trim-loss can be minimized. Although loading a pallet may seem linear to most people, some customers impose restrictions that increase the complexity of the pallet loading. Due to that, to evaluate the complexity of a packed pallet, some metrics were created. They consist in an evaluation of a set of parameters that are inherent to the pallet loading process and affect its complexity. After analysing some of those constraints and loading methods enforced by some pickers in a real company, it was possible to obtain samples where the metrics were applied to learn which parameters add the most complexity in the pallet loading process. In the future, after knowing the relevancy of each parameter, the metrics can be used in pallet generation tools to learn how complex is the loading of a certain pallet and study new and easier ways to load the boxes that reduce the complexity of such process. Two statistical tests were then used to analyse the samples retrieved: the principal components analysis and the multiple linear regression. The first is used to combine multiple variables into a smaller set that represents the most relevant information, while the multiple linear regression uses the variables and respective observations to calculate a model that can predict the value of the complexity of a packed pallet in given circumstances. In the first one, it was learned that three principal components were extracted, but since the third one explained a small percentage of the total data variance, it was decided to retain only two components: the box quantities, which explains 41% of the total variance, followed by the box dimensions, explaining 28% of the total variance. The multiple linear regression revealed that the component representing the box quantities, which contains the Number of Box Types, Number of Column Piles, Number of Boxes, Time Spent Packing, and Percentage of Fragile Boxes variables is the component that mostly increase the complexity of pallet cargo arrangements. Although the model can predict the data that was obtained with an average accuracy, some of the coefficients ended up being small, those being related to the components Box Dimensions, which has the Number of Heavy Boxes, Average Box Weight, Average Maximum Width variables, and Height Between Pile and Worker and Number variables, meaning that they aren’t very significant towards evaluating the complexity of a pallet loading process. Using a multiple linear regression with the 9 variables showed that the variable who adds more complexity is the Number of Column Piles. Overall, the results obtained were acceptable, and showed that the variable that adds more complexity is the ones that the pickers see as adding more complexity, and also that the results of the multiple regression with the components match the one using the original variables. It is worth noting that this variable is subjective, meaning that one worker’s perception on the complexity may not match others’ perception. Despite having obtained only one variable being considered as statistically significant towards explaining the complexity in the pallet loading problem, it doesn’t mean it’s the only one that adds complexity.No problema de carregamento de paletes, um dos grandes objetivos é alocar o maior número de caixas possível, visando minimizar espaços vazios conhecidos por trim-loss. Se todas as caixas possuírem um formato retangular, que é o formato mais comum, é possível arrumá-las de forma que as suas faces fiquem encostadas entre si, minimizando assim o trim-loss. No entanto, apesar do empacotamento de caixas em paletes parecer linear para a maioria das pessoas, certos clientes impõem restrições que aumentam a complexidade do empacotamento. Como tal, para avaliar a complexidade de um arranjo de paletes, criaram-se métricas, que consistem na avaliação de um conjunto de parâmetros inerentes ao processo ou às características do carregamento de paletes que afetam a sua complexidade. Após analisar numa empresa real as restrições e os métodos de empacotamento usados pelos operadores, foi possível obter amostras onde as métricas são aplicadas para tentar saber quais as mais relevantes no processo, para assim futuramente estas métricas serem aplicadas em ferramentas de geração de paletes para poder analisar os resultados obtidos e estudar maneiras onde estas sejam carregadas mais facilmente. Posteriormente, dois testes estatísticos foram aplicados aos dados recolhidos: uma análise de componentes principais e a regressão linear múltipla. O primeiro usa-se para combinar várias variáveis e formar um conjunto mais pequeno que represente a informação mais relevante, enquanto a regressão linear múltipla usa as variáveis e respetivas observações para calcular um modelo que consiga prever valores de complexidade do carregamento de paletes em quaisquer circunstâncias. No primeiro, verificou-se a existência de três componentes principais, mas dado que o terceiro componente explica uma percentagem da variância total dos dados pequena, decidiu-se extrair apenas dois componentes: as quantidades das caixas é o componente que explica maiores valores de variância nos dados (41%), seguido pelas dimensões das caixas, explicando 28% da variância total dos dados. A regressão linear múltipla revelou que o componente que representa as quantidades das caixas, que contém as variáveis Número de Tipos de Caixa, Número de Colunas, Número de Caixas, Tempo Despendido a Carregar Caixas e Percentagem de Caixas Frágeis, é aquele que faz crescer mais substancialmente a complexidade do carregamento de caixas em paletes. Com os vários testes, verificou-se que os componentes Dimensões das Caixas, que possui as variáveis Número de Caixas Pesadas Carregadas, Peso Médio das Caixas, Largura Máxima Média, e a diferença de alturas entre pilhas de caixas e o operador, não acrescentam muita significância na explicação da avaliação da complexidade no problema de carregamento de paletes. A regressão linear múltipla com as variáveis originais mostrou que o Número de Colunas é a variável que adiciona mais complexidade. Apesar do modelo obtido ter significância, quase todos os coeficientes obtidos acabaram por ser baixos e com valores Significância (sig.) acima de 0,05, não sendo essas variáveis relevantes no modelo. Valores baixos de Cronbach’s Alpha e R2 ajustado evidenciam a suscetibilidade da aparição destes valores. No geral, os resultados obtidos nesta dissertação foram satisfatórios, mas os coeficientes baixos da regressão linear múltipla não foram bons. O número de observações retido e o escalamento das variáveis são causas possíveis para esta discrepância de valores ter acontecido. Vale a pena referir que a variável que avalia a complexidade é uma variável subjetiva, pelo que o que um picker considera como sendo complexo pode não corresponder ao que outros trabalhadores pensem. Apesar de, estatisticamente, apenas uma variável ter significância na explicação da complexidade, na realidade todas as variáveis têm alguma influência na complexidade do carregamento de caixas em paletes. No geral, a perceção dos trabalhadores tem semelhanças com aquilo que se obteve nos resultados das regressões lineares

Learning Physically Realizable Skills for Online Packing of General 3D Shapes

We study the problem of learning online packing skills for irregular 3D shapes, which is arguably the most challenging setting of bin packing problems. The goal is to consecutively move a sequence of 3D objects with arbitrary shapes into a designated container with only partial observations of the object sequence. Meanwhile, we take physical realizability into account, involving physics dynamics and constraints of a placement. The packing policy should understand the 3D geometry of the object to be packed and make effective decisions to accommodate it in the container in a physically realizable way. We propose a Reinforcement Learning (RL) pipeline to learn the policy. The complex irregular geometry and imperfect object placement together lead to huge solution space. Direct training in such space is prohibitively data intensive. We instead propose a theoretically-provable method for candidate action generation to reduce the action space of RL and the learning burden. A parameterized policy is then learned to select the best placement from the candidates. Equipped with an efficient method of asynchronous RL acceleration and a data preparation process of simulation-ready training sequences, a mature packing policy can be trained in a physics-based environment within 48 hours. Through extensive evaluation on a variety of real-life shape datasets and comparisons with state-of-the-art baselines, we demonstrate that our method outperforms the best-performing baseline on all datasets by at least 12.8% in terms of packing utility.Comment: ACM Transactions on Graphics (TOG

Algorithms for the Bin Packing Problem with Scenarios

This paper presents theoretical and practical results for the bin packing problem with scenarios, a generalization of the classical bin packing problem which considers the presence of uncertain scenarios, of which only one is realized. For this problem, we propose an absolute approximation algorithm whose ratio is bounded by the square root of the number of scenarios times the approximation ratio for an algorithm for the vector bin packing problem. We also show how an asymptotic polynomial-time approximation scheme is derived when the number of scenarios is constant. As a practical study of the problem, we present a branch-and-price algorithm to solve an exponential model and a variable neighborhood search heuristic. To speed up the convergence of the exact algorithm, we also consider lower bounds based on dual feasible functions. Results of these algorithms show the competence of the branch-and-price in obtaining optimal solutions for about 59% of the instances considered, while the combined heuristic and branch-and-price optimally solved 62% of the instances considered

Volumetric Techniques for Product Routing and Loading Optimisation in Industry 4.0: A Review

Industry 4.0 has become a crucial part in the majority of processes, components, and related modelling, as well as predictive tools that allow a more efficient, automated and sustainable approach to industry. The availability of large quantities of data, and the advances in IoT, AI, and data-driven frameworks, have led to an enhanced data gathering, assessment, and extraction of actionable information, resulting in a better decision-making process. Product picking and its subsequent packing is an important area, and has drawn increasing attention for the research community. However, depending of the context, some of the related approaches tend to be either highly mathematical, or applied to a specific context. This article aims to provide a survey on the main methods, techniques, and frameworks relevant to product packing and to highlight the main properties and features that should be further investigated to ensure a more efficient and optimised approach