Exact solution approaches for bilevel lot-sizing

In this paper we propose exact solution methods for a bilevel uncapacitated lot-sizing problem with backlogs. This is an extension of the classical uncapacitated lot-sizing problem with backlogs, in which two autonomous and self-interested decision makers constitute a two-echelon supply chain. The leader buys items from the follower in order to meet external demand at lowest cost. The follower also tries to minimize its costs. Both parties may backlog. We study the leader's problem, i.e., how to determine supply requests over time to minimize its costs in view of the possible actions of the follower. We develop two mixed-integer linear programming reformulations, as well as cutting planes to cut off feasible, but suboptimal solutions. We compare the reformulations on a series of benchmark instances. © 2012 Elsevier B.V. All rights reserved

Multilevel Lot-Sizing with Inventory Bounds

We consider a single-item multilevel lot-sizing problem with a serial structure where one of the levels has an inventory capacity (the bottleneck level). We propose a novel dynamic programming algorithm combining Zangwill’s approach for the uncapacitated problem and the basis-path approach for the production capacitated problem. Under reasonable assumptions on the cost parameters the time complexity of the algorithm is O(LT6) with L the number of levels in the supply chain and T the length of the planning horizon. Computational tests show that our algorithm is significantly faster than the commercial solver CPLEX applied to a standard formulation and can solve reasonably sized instances up to 48 periods and 12 levels in a few minutes.</p

Mixed integer programming formulations and heuristics for joint production and transportation problems.

In this thesis we consider different joint production and transportation problems. We first study the simplest two-level problem, the uncapacitated two-level production-in-series lot-sizing problem (2L-S/LS-U). We give a new polynomial dynamic programming algorithm and a new compact extended formulation for the problem and for an extension with sales. Some computational tests are performed comparing several reformulations on a NP-Hard problem containing the 2L-S/LS-U as a relaxation. We also investigate the one-warehouse multi-retailer problem (OWMR), another NP-Hard extension of the 2L-S/LS-U. We study possible ways to tackle the problem effectively using mixed integer programming (MIP) techniques. We analyze the projection of a multi-commodity reformulation onto the space of the original variables for two special cases and characterize valid inequalities for the 2L-S/LS-U. Limited computational experiments are performed to compare several approaches. We then analyze a more general two-level production and transportation problem with multiple production sites. Relaxations for the problem for which reformulations are known are identified in order to improve the linear relaxation bounds. We show that some uncapacitated instances of the basic problem of reasonable size can often be solved to optimality. We also show that a hybrid MIP heuristic based on two different MIP formulations permits us to find solutions guaranteed to be within 10% of optimality for harder instances with limited transportation capacity and/or with additional sales. For instances with big bucket production or aggregate storage capacity constraints the gaps can be larger. In addition, we study a different type of production and transportation problem in which cllients place orders with different sizes and delivery dates and the transportation is performed by a third company. We develop a MIP formulation and an algorithm with a local search procedure that allows us to solve large instances effectively.

A polyhedral study of multiechelon lot sizing with intermediate demands

In this paper, we study a multiechelon uncapacitated lot-sizing problem in series (m-ULS), where the output of the intermediate echelons has its own external demand and is also an input to the next echelon. We propose a polynomial-time dynamic programming algorithm, which gives a tight, compact extended formulation for the two-echelon case (2-ULS). Next, we present a family of valid inequalities for m-ULS, show its strength, and give a polynomial-time separation algorithm. We establish a hierarchy between the alternative formulations for 2-ULS. In particular, we show that our valid inequalities can be obtained from the projection of the multicommodity formulation. Our computational results show that this extended formulation is very effective in solving our uncapacitated multi-item two-echelon test problems. In addition, for capacitated multi-item, multiechelon problems, we demonstrate the effectiveness of a branch-and-cut algorithm using the proposed inequalities. © 2012 INFORMS

Proposta de modelo matemático para a produção e teste de funcionalidade em ambiente de máquinas

Orientadora : Profª. Neida Maria Patias VolpiTese (doutorado) - Universidade Federal do Paraná, Setor de Tecnologia e Ciências Exatas, Programa de Pós-Graduação em Métodos Numéricos em Engenharia-Programação Matemática. Defesa: Curitiba, 10/04/2017Inclui referências : f. 104-109Resumo: Este trabalho está baseado nos estudos de modelos matemáticos para problemas de dimensionamento e sequenciamento de lotes existentes na literatura. Com base nesses trabalhos foi proposto um modelo matemático envolvendo um ambiente produtivo com máquinas distintas em paralelo em um setor de uma indústria que sincroniza dois estágios do ambiente operacional, ou seja, a produção e o teste de funcionalidade com estoque intermediário. O primeiro estágio consiste num ambiente onde ocorre a produção de itens em várias máquinas paralelas distintas. O segundo é formado por fornos de teste nos quais são simuladas situações climáticas adversas de modo a testar o funcionamento dos itens fabricados.É proposto um modelo matemático para auxílio no processo de gestão das ordens de produção, de modo a determinar em quais máquinas bem como em que ordem os itens serão produzidos no primeiro estágio, de modo a reduzir custos de produção neste estágio, bem como atender ao melhor aproveitamento possível dos fornos de teste, evitando a operação destes contendo espaços ociosos e controlando os estoques intermediários. É importante fazer um sincronismo na linha de produção durante o planejamento da produção, para evitar excessos de estoques e paradas de máquinas, reduzindo assim custos desnecessários. Devido à busca pela minimização de custos, torna-se necessário sempre que possível, utilizar ao máximo a capacidade dos fornos, buscando atender às demandas previstas para o horizonte de planejamento. Para validação do modelo, propõe-se uma aplicação com dados genéricos obtidos de uma empresa fabricante de inversores, o qual é resolvido com uso do software CPLEX e as heurísticas Relax-and-Fix e Fix-and-Optimize. Nos testes realizados a heurística Relax-and-Fix teve um desempenho melhor em cenários onde a quantidade de subproblemas foi maior. O modelo pode ser aplicado em outros ambientes industriais com características similares, como por exemplo, no processo de montagem de um computador ou de um refrigerador, entre outros. Palavras-chaves: Dimensionamento e Sequenciamento de Lotes. Estoques Intermediários. Teste de Funcionalidade.Abstract: This work is based on the studies of mathematical models for problems of dimensioning and sequencing of lots existing in the literature. Based on these works, a mathematical model involving a productive environment with distinct machines in parallel was proposed in a sector of an industry that synchronizes two stages of the operational environment, that is, the production and the test of functionality with intermediate stock. The first stage consists of an environment where the production of items occurs in several different parallel machines. The second is formed by test furnaces in which adverse weather conditions are simulated in order to test the operation of the manufactured items. A mathematical model is proposed to aid in the process of the management of the production orders, in order to determine in which machines as well as in which order the items will be produced in the first stage, in order to reduce production costs at this stage, as well as to take care of the best possible use of the test furnaces, avoiding the operation of these containing idle spaces and controlling the intermediate stocks. It is important to synchronize the production line during production planning to avoid overstocking and downtime, thus reducing unnecessary costs. Due to the search for cost minimization, it is necessary, whenever possible, to make maximum use of the furnace capacity, in order to meet the demands for the planning horizon. To validate the model, it is proposed an application with generic data obtained from an inverter manufacturer, which is solved using the CPLEX software and the Relax-and-Fix and Fix-and-Optimize heuristics. In the tests performed the Relax-and-Fix heuristic performed better in scenarios where the number of subproblems was greater. The model can be applied in other industrial environments with similar characteristics, for example, in the process of assembling a computer or a refrigerator, among others. Key-words: Lot Sizing and Scheduling. Intermediate Stocks. Functionality Test

Uncapacitated two-level lot-sizing

For the uncapacitated two-level production-in-series T period lot-sizing model, a dynamic program with running time O(T^2 log T) and a compact and tight extended formulation with O(T^3) variables and O(T^2) equality constraints are presented. Limited computational comparisons of various formulations of two level production/transportation problems with multiple clients are reported