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

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

A theoretical study of two-period relaxations for lot-sizing problems with big-bucket capacities

In this paper, we study two-period subproblems proposed by Akartunali et al. (2015) for lot-sizing problems with big-bucket capacities and nonzero setup times, complementing our previous work investigating the special case of zero setup times. In particular, we study the polyhedral structure of the mixed integer sets related to various two-period relaxations. We derive several families of valid inequalities and investigate their facet-defining conditions. We also discuss the separation problems associated with these valid inequalities

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

The Minimum Cost Design of Transparent Optical Networks Combining Grooming, Routing, and Wavelength Assignment

As client demands grow, optical network operators are required to introduce lightpaths of higher line rates in order to groom more demand into their network capacity. For a given fiber network and a given set of client demands, the minimum cost network design is the task of assigning routing paths and wavelengths for a minimum cost set of lightpaths able to groom all client demands. The variant of the optical network design problem addressed in this paper considers a transparent optical network, single hop grooming, client demands of a single interface type, and lightpaths of two line rates. We discuss two slightly different mixed integer linear programming models that define the network design problem combining grooming, routing, and wavelength assignment. Then, we propose a parameters increase rule and three types of additional constraints that, when applied to the previous models, make their linear relaxation solutions closer to the integer solutions. Finally, we use the resulting models to derive a hybrid heuristic method, which combines a relax-and-fix approach with an integer linear programming-based local search approach. We present the computational results showing that the proposed heuristic method is able to find solutions with cost values very close to the optimal ones for a real nation-wide network and considering a realistic fiber link capacity of 80 wavelengths. Moreover, when compared with other approaches used in the problem variants close to the one addressed here, our heuristic is shown to compute solutions, on average, with better cost values and/or in shorter runtimes

Estimating the impact of lifestyle changes on treatment outcomes for people with knee osteoarthritis through system dynamics simulation modelling

With the increasing number of patients suffering from knee osteoarthritis, the UK National Health Service is considering introducing a new treatment option that would focus on lifestyle changes. This study aims to develop a novel model that could serve as a tool to estimate the impact of such an intervention on treatment outcomes. In collaboration with the Forth Valley Royal Hospital, Larbert, United Kingdom, the model was formulated as a system dynamics simulation model and was built using Insight Maker, a web-based modelling tool. To the best of our knowledge, this paper is the first to employ system dynamics to tackle this problem. The simulations were run for several confgurations to better understand the potential impact of advanced lifestyle treatment under various scenarios. The results for the most expected scenario suggest that introducing advanced lifestyle treatment would increase the average number of recovered patients by 4%, and reduce the average numbers of temporarily disabled, permanently disabled and deceased patients by 21%, 9% and 4%, respectively. The results also reveal that even with low advanced lifestyle treatment acceptance rates, the treatment outcomes could improve without any changes to current resources

Valid inequalities for a single constrained 0-1 MIP set intersected with a conflict graph

In this paper a mixed integer set resulting from the intersection of a single constrained mixed 0–1 set with the vertex packing set is investigated. This set arises as a subproblem of more general mixed integer problems such as inventory routing and facility location problems. Families of strong valid inequalities that take into account the structure of the simple mixed integer set and that of the vertex packing set simultaneously are introduced. In particular, the well-known mixed integer rounding inequality is generalized to the case where incompatibilities between binary variables are present. Exact and heuristic algorithms are designed to solve the separation problems associated to the proposed valid inequalities. Preliminary computational experiments show that these inequalities can be useful to reduce the integrality gaps and to solve integer programming problems

MOMO - multi-objective metabolic mixed integer optimization : application to yeast strain engineering

BACKGROUND: In this paper, we explore the concept of multi-objective optimization in the field of metabolic engineering when both continuous and integer decision variables are involved in the model. In particular, we propose a multi-objective model that may be used to suggest reaction deletions that maximize and/or minimize several functions simultaneously. The applications may include, among others, the concurrent maximization of a bioproduct and of biomass, or maximization of a bioproduct while minimizing the formation of a given by-product, two common requirements in microbial metabolic engineering. RESULTS: Production of ethanol by the widely used cell factory Saccharomyces cerevisiae was adopted as a case study to demonstrate the usefulness of the proposed approach in identifying genetic manipulations that improve productivity and yield of this economically highly relevant bioproduct. We did an in vivo validation and we could show that some of the predicted deletions exhibit increased ethanol levels in comparison with the wild-type strain. CONCLUSIONS: The multi-objective programming framework we developed, called MOMO, is open-source and uses POLYSCIP (Available at http://polyscip.zib.de/). as underlying multi-objective solver. MOMO is available at http://momo-sysbio.gforge.inria.fr

A theoretical and computational analysis of lot-sizing in remanufacturing with separate setups

Due to the stricter government regulations on end-of-life product treatment and the increasing public awareness towards environmental issues, remanufacturing has been a significantly growing industry over the last decades, offering many potential business opportunities. In this paper, we investigate a crucial problem apparent in this industry, the remanufacturing lot-sizing problem with separate setups. We first discuss two reformulations of this problem, and remark an important property with regards to their equivalence. Then, we present a theoretical investigation of a related subproblem, where our analysis indicates that a number of flow cover inequalities are strong for this subproblem under some general conditions. We then investigate the computational effectiveness of the alternative methods discussed for the original problem. Detailed numerical results are insightful for the practitioner, indicating that in particular when the return variability increases or when the remanufacturing setup costs decrease relevant to manufacturing setup costs, the flow covers can be very effective

MOMO - multi-objective metabolic mixed integer optimization : application to yeast strain engineering

In this paper, we explore the concept of multi-objective optimization in the field of metabolic engineering when both continuous and integer decision variables are involved in the model. In particular, we propose a multi-objective model that may be used to suggest reaction deletions that maximize and/or minimize several functions simultaneously. The applications may include, among others, the concurrent maximization of a bioproduct and of biomass, or maximization of a bioproduct while minimizing the formation of a given by-product, two common requirements in microbial metabolic engineering. Production of ethanol by the widely used cell factory Saccharomyces cerevisiae was adopted as a case study to demonstrate the usefulness of the proposed approach in identifying genetic manipulations that improve productivity and yield of this economically highly relevant bioproduct. We did an in vivo validation and we could show that some of the predicted deletions exhibit increased ethanol levels in comparison with the wild-type strain. The multi-objective programming framework we developed, called Momo, is open-source and uses PolySCIP‡ as underlying multi-objective solver. Momo is available at http://momo-sysbio.gforge.inria.f