Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems

In this thesis, we provide (or review) new and effective algorithms based on Mixed-Integer Linear Programming (MILP) models and/or decomposition approaches to solve exactly various cutting and packing problems. The first three contributions deal with the classical bin packing and cutting stock problems. First, we propose a survey on the problems, in which we review more than 150 references, implement and computationally test the most common methods used to solve the problems (including branch-and-price, constraint programming (CP) and MILP), and we successfully propose new instances that are difficult to solve in practice. Then, we introduce the BPPLIB, a collection of codes, benchmarks, and links for the two problems. Finally, we study in details the main MILP formulations that have been proposed for the problems, we provide a clear picture of the dominance and equivalence relations that exist among them, and we introduce reflect, a new pseudo-polynomial formulation that achieves state of the art results for both problems and some variants. The following three contributions deal with two-dimensional packing problems. First, we propose a method using Logic based Benders’ decomposition for the orthogonal stock cutting problem and some extensions. We solve the master problem through an MILP model while CP is used to solve the slave problem. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. Then, we introduce TwoBinGame, a visual application we developed for students to interactively solve two-dimensional packing problems, and analyze the results obtained by 200 students. Finally, we study a complex optimization problem that originates from the packaging industry, which combines cutting and scheduling decisions. For its solution, we propose mathematical models and heuristic algorithms that involve a non-trivial decomposition method. In the last contribution, we study and strengthen various MILP and CP approaches for three project scheduling problems

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes

Models and algorithms for hard optimization problems

This thesis is devoted to exact solution methods for NP-hard integer programming models. We consider two of these problems, the cutting stock problem and the vehicle routing problem. Both problems have been studied for several decades by researchers and practitioners of the Operations Research eld. Their interest and contribution to real-world applications in business, industry and several kinds of organizations are irrefutable. Our solution approaches are always exact. We contribute with new lower bounds, families of valid inequalities, integer programming models and exact algorithms for the problems we explore. More precisely, we address two variants of each of the referred problems. In what concerns cutting stock problems, we analyze the one-dimensional pattern minimization problem and the two-dimensional cutting stock problem with the guillotine constraint. The one-dimensional pattern minimization problem is a cutting and packing problem that becomes relevant in situations where changing from one pattern to another involves, for example, a cost for setting up the cutting machine. It is the problem of minimizing the number of di erent patterns of a given cutting stock solution. For this problem, we contribute with new lower bounds. The two-dimensional cutting stock problem with the guillotine constraint and two stages is also addressed. We propose a pseudo-polynomial network ow model, along with some reduction criteria to reduce its symmetry. We strengthen the model with a new family of cutting planes and propose a new lower bound. For this variant, we also consider some variations of the problem.Regarding vehicle routing problems, we address the vehicle routing problem with time windows and multiple use of vehicles and the location routing problem, with capacitated vehicles and depots and multiple use of vehicles. The rst of these problems considers the well know case of vehicle routing with time windows with the additional consideration that vehicles can be assigned to several routes within the same planning period. The second variant considers the combination of the rst problem, without time windows, with a location problem. This means that the depots to be used must be selected from a set of available ones. For both of these variants, we propose a network ow model whose nodes of the underlying graph correspond to time instants of the planning period and whose arcs correspond to vehicle routes. We reduce their symmetry by deriving several reduction criteria. For the vehicle routing problem with time windows and multiple use of vehicles, we propose an iterative algorithm to solve the problem exactly. Our proposed procedures are tested and compared with other methods from the literature. All the computational results produced by the series of experiments are presented and discussed.Esta tese e dedicada a métodos de resolução exata para problemas de programação inteira NP-difíceis. São considerados dois desses problemas, nomeadamente o problema de corte e empacotamento e o problema de encaminhamento de veículos. Ambos os problemas têm vindo a ser abordados por investigadores e profissionais da área da Investigação Operacional há já várias décadas. O seu interesse e contribuição para aplicações reais do mundo dos negócios e industria, assim como para inúmeros outros tipos de organizações são, hoje em dia, inegáveis. A nossa abordagem para a resolução dos problemas descritos e exata. Contribuímos com novos limites inferiores, novas famílias de desigualdades validas, novos modelos de programação inteira e algoritmos de resolução exata para os problemas que nos propomos explorar. Em particular, abordamos duas variantes de cada um dos referidos problemas. Em relação ao problema de corte e empacotamento, analisamos o problema de minimização de padrões a uma dimensão e o problema de corte e empacotamento a duas dimensões, com restrição de guilhotina. O problema de minimização de padrões a uma dimensão e pertinente em situações em que a mudança de padrão envolve, por exemplo, custos de reconfiguração nas máquinas de corte. E o problema de minimização do numero de padrões diferentes de uma dada solução de um problema de corte. Para este problema contribuímos com novos limites inferiores. O problema de corte e empacotamento a duas dimensões com restrição de guilhotina e dois estágios e também abordado. Propomos um modelo pseudopolinomial de rede de fluxos, assim como critérios de redução que eliminam parte da sua simetria. Reforçamos o modelo com uma nova família de planos de corte e propomos novos limites inferiores. Para esta variante, consideramos também outras variações do problema original. No que se refere ao problema de encaminhamento de veículos, abordamos um problema de encaminhamento de veículos com janelas temporais e múltiplas viagens, e também um problema de localização e encaminhamento de veículos com capacidades nos veículos e depósitos e múltiplo uso dos veículos. O primeiro destes problemas considera o conhecido caso de encaminhamento de veículos com janelas temporais, com a consideração adicional de que os veículos podem ser alocados a v arias rotas no decurso do mesmo período de planeamento. A segunda variante considera a combinação do primeiro problema, embora sem janelas temporais, com um problema de localização. Isto significa que os depósitos a usar são selecionados de um conjunto de localizações disponíveis. Para ambas as variantes, propomos um modelo pseudo-polinomial de rede de fluxos cujos nodos do grafo correspondente representam instantes de tempo do período de planeamento, e cujos arcos representam rotas. Derivamos critérios de redução com o intuito de reduzir a simetria. Para o problema com janelas temporais e múltiplas viagens, propomos um algoritmo iterativo que o resolve de forma exata. Os procedimentos propostos são testados e comparados com outros métodos da literatura. Todos os resultados obtidos pelas experiencias computacionais são apresentados e discutidos

Algorithms for vehicle routing problems with heterogeneous fleet, flexible time windows and stochastic travel times

Orientador: Vinícius Amaral ArmentanoTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Este trabalho aborda três variantes multiatributo do problema de roteamento de veículos. A primeira apresenta frota heterogênea, janelas de tempo invioláveis e tempos de viagem determinísticos. Para resolvê-la, são propostos algoritmos ótimos baseados na decomposição de Benders. Estes algoritmos exploram a estrutura do problema em uma formulação de programação inteira mista, e três diferentes técnicas são desenvolvidas para acelerá-los. A segunda variante contempla os atributos de frota heterogênea, janelas de tempo flexíveis e tempos de viagem determinísticos. As janelas de tempo flexíveis permitem o início do serviço nos clientes com antecipação ou atraso limitados em relação às janelas de tempo invioláveis, com custos de penalidade. Este problema é resolvido por extensões dos algoritmos de Benders, que incluem novos algoritmos de programação dinâmica para a resolução de subproblemas com a estrutura do problema do caixeiro viajante com janelas de tempo flexíveis. A terceira variante apresenta frota heterogênea, janelas de tempo flexíveis e tempos de viagem estocásticos, sendo representada por uma formulação de programação estocástica inteira mista de dois estágios com recurso. Os tempos de viagem estocásticos são aproximados por um conjunto finito de cenários, gerados por um algoritmo que os descreve por meio da distribuição de probabilidade Burr tipo XII, e uma matheurística de busca local granular é sugerida para a resolução do problema. Extensivos testes computacionais são realizados em instâncias da literatura, e as vantagens das janelas de tempo flexíveis e dos tempos de viagem estocásticos são enfatizadasAbstract: This work addresses three multi-attribute variants of the vehicle routing problem. The first one presents a heterogeneous fleet, hard time windows and deterministic travel times. To solve this problem, optimal algorithms based on the Benders decomposition are proposed. Such algorithms exploit the structure of the problem in a mixed-integer programming formulation, and three algorithmic enhancements are developed to accelerate them. The second variant comprises a heterogeneous fleet, flexible time windows and deterministic travel times. The flexible time windows allow limited early and late servicing at customers with respect to their hard time windows, at the expense of penalty costs. This problem is solved by extensions of the Benders algorithms, which include novel dynamic programming algorithms for the subproblems with the special structure of the traveling salesman problem with flexible time windows. The third variant presents a heterogeneous fleet, flexible time windows and stochastic travel times, and is represented by a two-stage stochastic mixed-integer programming formulation with recourse. The stochastic travel times are approximated by a finite set of scenarios generated by an algorithm which describes them using the Burr type XII distribution, and a granular local search matheuristic is suggested to solve the problem. Extensive computational tests are performed on instances from the literature, and the advantages of flexible windows and stochastic travel times are stressed.DoutoradoAutomaçãoDoutor em Engenharia Elétrica141064/2015-3CNP

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order

Operational research:methods and applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order