An Improved Arcflow Model for the Skiving Stock Problem

Because of the sharp development of (commercial) MILP software and hardware components, pseudo-polynomial formulations have been established as a viable tool for solving cutting and packing problems in recent years. Constituting a natural (but independent) counterpart of the well-known cutting stock problem, the one-dimensional skiving stock problem (SSP) asks for the maximal number of large objects (specified by some threshold length) that can be obtained by recomposing a given inventory of smaller items. In this paper, we introduce a new arcflow formulation for the SSP applying the idea of reflected arcs. In particular, this new model is shown to possess significantly fewer variables as well as a better numerical performance compared to the standard arcflow formulation

Improved flow-based formulations for the skiving stock problem

Thanks to the rapidly advancing development of (commercial) MILP software and hardware components, pseudo-polynomial formulations have been established as a powerful tool for solving cutting and packing problems in recent years. In this paper, we focus on the one-dimensional skiving stock problem (SSP), where a given inventory of small items has to be recomposed to obtain a maximum number of larger objects, each satisfying a minimum threshold length. In the literature, different modeling approaches for the SSP have been proposed, and the standard flow-based formulation has turned out to lead to the best trade-off between efficiency and solution time. However, especially for instances of practically meaningful sizes, the resulting models involve very large numbers of variables and constraints, so that appropriate reduction techniques are required to decrease the numerical efforts. For that reason, this paper introduces two improved flow-based formulations for the skiving stock problem that are able to cope with much larger problem sizes. By means of extensive experiments, these new models are shown to possess significantly fewer variables as well as an average better computational performance compared to the standard arcflow formulation

Logic based Benders' decomposition for orthogonal stock cutting problems

We consider the problem of packing a set of rectangular items into a strip of fixed width, without overlapping, using minimum height. Items must be packed with their edges parallel to those of the strip, but rotation by 90\ub0 is allowed. The problem is usually solved through branch-and-bound algorithms. We propose an alternative method, based on Benders' decomposition. The master problem is solved through a new ILP model based on the arc flow formulation, while constraint programming is used to solve the slave problem. The resulting method is hybridized with a state-of-the-art branch-and-bound algorithm. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. We additionally show that the algorithm can be successfully used to solve relevant related problems, like rectangle packing and pallet loading

A combinatorial flow-based formulation for temporal bin packing problems

We consider two neighboring generalizations of the classical bin packing problem: the temporal bin packing problem (TBPP) and the temporal bin packing problem with fire-ups (TBPP-FU). In both cases, the task is to arrange a set of given jobs, characterized by a resource consumption and an activity window, on homogeneous servers of limited capacity. To keep operational costs but also energy consumption low, TBPP is concerned with minimizing the number of servers in use, whereas TBPP-FU additionally takes into account the switch-on processes required for their operation. Either way, challenging integer optimization problems are obtained, which can differ significantly from each other despite the seemingly only marginal variation of the problems. In the literature, a branch-and-price method enriched with many preprocessing steps (for TBPP) and compact formulations (for TBPP-FU), benefiting from numerous reduction methods, have emerged as, currently, the most promising solution methods. In this paper, we introduce, in a sense, a unified solution framework for both problems (and, in fact, a wide variety of further interval scheduling applications) based on graph theory. Any scientific contributions in this direction failed so far because of the exponential size of the associated networks. The approach we present in this article does not change the theoretical exponentiality itself, but it can make it controllable by clever construction of the resulting graphs. In particular, for the first time all classical benchmark instances (and even larger ones) for the two problems can be solved – in times that significantly improve those of the previous approaches

Pattern Generation for Three Dimensional Cutting Stock Problem

We consider the problem of three-dimensional cutting of a large block that is to be cut into some small block pieces, each with a specific size and request. Pattern generation is an algorithm that has been used to determine cutting patterns in one-dimensional and two-dimensional problems. The purpose of this study is to modify the pattern generation algorithm so that it can be used in three-dimensional problems, and can determine the cutting pattern with the minimum possible cutting residue. The large block will be cut based on the length, width, and height. The rest of the cuts will be cut back if possible to minimize the rest. For three-dimensional problems, we consider the variant in which orthogonal rotation is allowed. By allowing the remainder of the initial cut to be rotated, the dimensions will have six permutations. The result of the calculation using the pattern generation algorithm for three-dimensional problems is that all possible cutting patterns are obtained but there are repetitive patterns because they suggest the same number of cuts.

Arc flow formulations based on dynamic programming: Theoretical foundations and applications

Network flow formulations are among the most successful tools to solve optimization problems. Such formulations correspond to determining an optimal flow in a network. One particular class of network flow formulations is the arc flow, where variables represent flows on individual arcs of the network. For NP-hard problems, polynomial-sized arc flow models typically provide weak linear relaxations and may have too much symmetry to be efficient in practice. Instead, arc flow models with a pseudo-polynomial size usually provide strong relaxations and are efficient in practice. The interest in pseudo-polynomial arc flow formulations has grown considerably in the last twenty years, in which they have been used to solve many open instances of hard problems. A remarkable advantage of pseudo-polynomial arc flow models is the possibility to solve practical-sized instances directly by a Mixed Integer Linear Programming solver, avoiding the implementation of complex methods based on column generation. In this survey, we present theoretical foundations of pseudo-polynomial arc flow formulations, by showing a relation between their network and Dynamic Programming (DP). This relation allows a better understanding of the strength of these formulations, through a link with models obtained by Dantzig-Wolfe decomposition. The relation with DP also allows a new perspective to relate state-space relaxation methods for DP with arc flow models. We also present a dual point of view to contrast the linear relaxation of arc flow models with that of models based on paths and cycles. To conclude, we review the main solution methods and applications of arc flow models based on DP in several domains such as cutting, packing, scheduling, and routing

Mathematical models and decomposition methods for the multiple knapsack problem

We consider the multiple knapsack problem, that calls for the optimal assignment of a set of items, each having a profit and a weight, to a set of knapsacks, each having a maximum capacity. The problem has relevant managerial implications and is known to be very difficult to solve in practice for instances of realistic size. We review the main results from the literature, including a classical mathematical model and a number of improvement techniques. We then present two new pseudo-polynomial formulations, together with specifically tailored decomposition algorithms to tackle the practical difficulty of the problem. Extensive computational experiments show the effectiveness of the proposed approaches

2차원 2단계 배낭문제에 대한 정수계획모형 및 최적해법

학위논문 (석사) -- 서울대학교 대학원 : 공과대학 산업공학과, 2021. 2. 이경식.In this thesis, we study integer programming models and exact algorithms for the two-dimensional two-staged knapsack problems, which maximizes the profit by cutting a single rectangular plate into smaller rectangular items by two-staged guillotine cuts. We first introduce various integer programming models, including the strip-packing model, the staged-pattern model, the level-packing model, and the arc-flow model for the problem. Then, a hierarchy of the strength of the upper bounds provided by the LP-relaxations of the models is established based on theoretical analysis. We also show that there exists a polynomial-size model that has not been proven yet as far as we know. Exact methods, including branch-and-price algorithms using the strip-packing model and the staged-pattern model, are also devised. Computational experiments on benchmark instances are conducted to examine the strength of upper bounds obtained by the LP-relaxations of the models and evaluate the performance of exact methods. The results show that the staged-pattern model gives a competitive theoretical and computational performance.본 논문은 2단계 길로틴 절단(two-staged guillotine cut)을 사용하여 이윤을 최대화하는 2차원 2단계 배낭 문제(two-dimensional two-staged knapsack problem: 이하 2TDK)에 대한 정수최적화 모형과 최적해법을 다룬다. 우선, 본 연구에서는 스트립패킹모형, 단계패턴모형, 레벨패킹모형, 그리고 호-흐름모형과 같은 정수최적화 모형들을 소개한다. 그 뒤, 각각의 모형의 선형계획완화문제에 대해 상한강도를 이론적으로 분석하여 상한강도 관점에서 모형들 간 위계를 정립한다. 또한, 본 연구에서는 2TDK의 다항크기(polynomial-size) 모형의 존재성을 처음으로 증명한다. 다음으로 본 연구는 2TDK의 최적해를 구하는 알고리즘으로써 패턴기반모형들에 대한 분지평가 알고리즘과 레벨패킹모형을 기반으로 한 분지절단 알고리즘을 제안한다. 단계패턴모형이 이론적으로도 가장 좋은 상한강도를 보장할 뿐만 아니라, 계산 분석을 통해 단계패턴모형을 기반으로 한 분지평가 알고리즘이 제한된 시간 내 좋은 품질의 가능해를 찾음을 확인하였다.Abstract i Contents iv List of Tables vi List of Figures vii Chapter 1 Introduction 1 1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 2 Integer Programming Models for 2TDK 9 2.1 Pattern-based Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Arc-flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Level Packing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Chapter 3 Theoretical Analysis of Integer Programming Models 20 3.1 Upper Bounds of AF and SM(1;1) . . . . . . . . . . . . . . . . . . 20 3.2 Upper Bounds of ML, PM(d), and SM(d; d) . . . . . . . . . . . . . . 21 3.3 Polynomial-size Model . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter 4 Exact Methods 33 4.1 Branch-and-price Algorithm for the Strip Packing Model . . . . . . . 34 4.2 Branch-and-price Algorithm for the Staged-pattern Model . . . . . . 39 4.2.1 The Standard Scheme . . . . . . . . . . . . . . . . . . . . . . 39 4.2.2 The Height-aggregated Scheme . . . . . . . . . . . . . . . . . 40 4.3 Branch-and-cut Algorithm for the Modified Level Packing Model . . 44 Chapter 5 Computational Experiments 46 5.1 Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2 Upper Bounds Comparison . . . . . . . . . . . . . . . . . . . . . . . 49 5.2.1 A Group of Small Instances . . . . . . . . . . . . . . . . . . . 49 5.2.2 A Group of Large Instances . . . . . . . . . . . . . . . . . . . 55 5.2.3 Class 5 Instances . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3 Solving Instances to Optimality . . . . . . . . . . . . . . . . . . . . . 65 5.3.1 A Group of Small Instances . . . . . . . . . . . . . . . . . . . 65 5.3.2 A Group of Large Instances . . . . . . . . . . . . . . . . . . . 69 5.3.3 Class 5 Instances . . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter 6 Conclusion 77 Bibliography 79 국문초록 83Maste

Problema generalizado del empaquetamiento de contenedores: una comparación entre diferentes métodos de solución

Objetivos y método de estudio: ´ El objetivo general de esta tesis es analizar modelos de propagación de restricciones para el problema generalizado del empaquetamiento de contenedores. El problema generalizado de empaquetamiento de contenedores consta de dos clases de objetos que deben ser empacados, los obligatorios y los no obligatorios. Cada objeto tiene un beneficio por ser empaquetado y un volumen. Los contenedores tienen un costo por uso y una capacidad que no debe ser excedida. El modelo de flujo en redes es uno de los mejores métodos resolviendo los problemas Bin Packing. Por ello utilizaremos este modelo para evaluar el desempeño de los modelos a ser resueltos por propagación de restricciones