Valid inequalities for the single arc design problem with set-ups

We consider a mixed integer set which generalizes two well-known sets: the single node fixed-charge network set and the single arc design set. Such set arises as a relaxation of feasible sets of general mixed integer problems such as lot-sizing and network design problems. We derive several families of valid inequalities that, in particular, generalize the arc residual capacity inequalities and the flow cover inequalities. For the constant capacitated case we provide an extended compact formulation and give a partial description of the convex hull in the original space which is exact under a certain condition. By lifting some basic inequalities we provide some insight on the difficulty of obtaining such a full polyhedral description for the constant capacitated case. Preliminary computational results are presented

Meta-Heuristics for Dynamic Lot Sizing: a review and comparison of solution approaches

Proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve. Because these problems are so difficult, various solution techniques have been proposed to solve them. In the past decade, meta-heuristics such as tabu search, genetic algorithms and simulated annealing, have become popular and efficient tools for solving hard combinational optimization problems. We review the various meta-heuristics that have been specifically developed to solve lot sizing problems, discussing their main components such as representation, evaluation neighborhood definition and genetic operators. Further, we briefly review other solution approaches, such as dynamic programming, cutting planes, Dantzig-Wolfe decomposition, Lagrange relaxation and dedicated heuristics. This allows us to compare these techniques. Understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms. Finally, we discuss general guidelines for computational experiments and illustrate these with several examples

Modeling Industrial Lot Sizing Problems: A Review

In this paper we give an overview of recent developments in the field of modeling single-level dynamic lot sizing problems. The focus of this paper is on the modeling various industrial extensions and not on the solution approaches. The timeliness of such a review stems from the growing industry need to solve more realistic and comprehensive production planning problems. First, several different basic lot sizing problems are defined. Many extensions of these problems have been proposed and the research basically expands in two opposite directions. The first line of research focuses on modeling the operational aspects in more detail. The discussion is organized around five aspects: the set ups, the characteristics of the production process, the inventory, demand side and rolling horizon. The second direction is towards more tactical and strategic models in which the lot sizing problem is a core substructure, such as integrated production-distribution planning or supplier selection. Recent advances in both directions are discussed. Finally, we give some concluding remarks and point out interesting areas for future research

Automated control system for the process of managing the procurement of raw materials used in steel making

The article presents a solution to the important problem of developing an automated control system for the process of purchasing raw materials used in metallurgical production. The system is based on an integrated approach that offers the solution that enables the performance of two main tasks: identifying the optimal lot size ordering and selecting a qualified supplier

Automated control system for the process of managing the procurement of raw materials used in steel making

The article presents a solution to the important problem of developing an automated control system for the process of purchasing raw materials used in metallurgical production. The system is based on an integrated approach that offers the solution that enables the performance of two main tasks: identifying the optimal lot size ordering and selecting a qualified supplier

Dynamic lot sizing with multiple suppliers, backlogging and quantity discounts

This paper studies the dynamic lot sizing problem with supplier selection, backlogging and quantity discounts. Two known discount types are considered separately, incremental and all-units quantity discounts. Mixed integer linear programming (MILP) formulations are presented for each case and solved using a commercial optimization software. In order to timely solve the problem, a recursive formulation and its efficient implementation are introduced for each case which result in an optimal and a near optimal solution for incremental and all-units quantity discount cases, respectively. Finally, the execution times of the MILP models and forward dynamic programming models obtained from the recursive formulations are presented and compared. The results demonstrate the efficiency of the dynamic programming models, as they can solve even large-sized instances quite timely. © 201

Estudo poliédrico de modelos de programação inteira mista que ocorrem em problemas de lot-sizing

Doutoramento conjunto em Matemática - Matemática e Aplicações (PDMA)“Branch-and-cut” algorithm is one of the most efficient exact approaches to solve mixed integer programs. This algorithm combines the advantages of a pure branch-and-bound approach and cutting planes scheme. Branch-and-cut algorithm computes the linear programming relaxation of the problem at each node of the search tree which is improved by the use of cuts, i.e. by the inclusion of valid inequalities. It should be taken into account that selection of strongest cuts is crucial for their effective use in branch-and-cut algorithm. In this thesis, we focus on the derivation and use of cutting planes to solve general mixed integer problems, and in particular inventory problems combined with other problems such as distribution, supplier selection, vehicle routing, etc. In order to achieve this goal, we first consider substructures (relaxations) of such problems which are obtained by the coherent loss of information. The polyhedral structure of those simpler mixed integer sets is studied to derive strong valid inequalities. Finally those strong inequalities are included in the cutting plane algorithms to solve the general mixed integer problems. We study three mixed integer sets in this dissertation. The first two mixed integer sets arise as a subproblem of the lot-sizing with supplier selection, the network design and the vendor-managed inventory routing problems. These sets are variants of the well-known single node fixed-charge network set where a binary or integer variable is associated with the node. The third set occurs as a subproblem of mixed integer sets where incompatibility between binary variables is considered. We generate families of valid inequalities for those sets, identify classes of facet-defining inequalities, and discuss the separation problems associated with the inequalities. Then cutting plane frameworks are implemented to solve some mixed integer programs. Preliminary computational experiments are presented in this direction.O algoritmo “branch-and-cut” é um dos métodos exatos mais eficientes para resolver problemas de programação inteira mista. Este algoritmo combina as vantagens do algoritmo branch-and-bound com o método de planos de corte. O algoritmo branch-and-cut recorre ao cálculo da relaxação linear em cada nó da árvore de pesquisa, a qual é melhorada com a utilização de cortes, isto é, com a inclusão de desigualdades válidas. Deve-se ter em conta que a escolha dos cortes mais fortes é crucial para a sua utilização efetiva no algoritmo branch-and-cut. Esta tese centra-se na obtenção de desigualdades válidas e sua utilização como planos de corte para resolver problemas gerais de programação inteira mista, em particular, problemas que combinam a gestão de stocks com outros problemas, tais como: a distribuição, selecção de fornecedores, e determinação de rotas de veículos, etc. Para alcançar este objetivo, são consideradas, em primeiro lugar, subestruturas, isto é, modelos de programação inteira mista que definem conjuntos de soluções admissíveis resultantes de relaxações desses problemas gerais. A estrutura poliédrica desses modelos é estudada de modo a serem obtidas novas famílias de desigualdades válidas. Finalmente, essas desigualdades são incluídas em algoritmos de planos de corte para resolver os problemas gerais de programação inteira mista. Nesta dissertação estudamos três modelos de programação inteira mista. Os dois primeiros modelos surgem como relaxações de problemas gerais tais como: dimensionamento de lotes com seleção de fornecedores, desenho de redes, e problemas que combinam a produção com a distribuição. Esses conjuntos constituem variantes do conhecido single node fixed-charge network set, onde uma variável binária ou inteira está associada a cada nó. O terceiro modelo ocorre como relaxação de problemas de programação inteira mista onde são consideradas incompatibilidades entre pares de variáveis binárias. Para os três modelos são geradas famílias de desigualdades válidas, são identificadas classes de desigualdades que definem facetas, e são discutidos os problemas de separação associados a essas desigualdades. Em seguida, essas desigualdades são utilizadas em algoritmos de planos de corte. É apresentada uma experiência computacional preliminar