Valid inequalities and projecting the multicommodity extended formulation for uncapacitated fixed charge network flow problems

Rardin, Ronald L.; Wolsey, Laurence A.

doi:10.1016/0377-2217(93)90263-m

Cited by 65 publications

(53 citation statements)

References 8 publications

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“…An alternative proof can be obtained by using the dicut collection inequalities of Rardin and Wolsey (1993). We provide the precise correspondence between the simple dicut collection inequalities and the 2-echelon inequalities in Corollary 9.…”

Section: Valid Inequalitiesmentioning

confidence: 96%

“…The formulation obtained by adding the projection inequalities (32) corresponding to the non-negative extreme rays with equal positive entries has the same strength as the formulation obtained by adding all 2-echelon inequalities (16). Rardin and Wolsey (1993) give a class of dicut collection inequalities for single-source uncapacitated fixed-charge networks, which are obtained by projecting the multi-commodity extended formulation to the original space. Dicut collection inequalities are written implicitly as a function of a collection of dicuts in a graph.…”

Section: Alternative Extended Formulations For 2-ulsmentioning

confidence: 99%

“…We refer the reader to Rardin and Wolsey (1993) for further details on the dicut collection inequalities.…”

Section: Alternative Extended Formulations For 2-ulsmentioning

confidence: 99%

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A Polyhedral Study of Multiechelon Lot Sizing with Intermediate Demands

Zhang

Küçükyavuz

Yaman

2012

Operations Research

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Abstract. In this paper, we study a multi-echelon 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 multi-commodity formulation. Our computational results show that this extended formulation is very effective in solving our uncapacitated multi-item 2-echelon test problems. In addition, for capacitated multiitem multi-echelon problems, we demonstrate the effectiveness of a branch-and-cut algorithm using the proposed inequalities.

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Section: Valid Inequalitiesmentioning

confidence: 96%

Section: Alternative Extended Formulations For 2-ulsmentioning

confidence: 99%

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A Polyhedral Study of Multiechelon Lot Sizing with Intermediate Demands

Zhang

Küçükyavuz

Yaman

2012

Operations Research

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“…Um modelo de programação matemática inteira mista típica para o problema (Rardin & Wolsey, 1993) definida sobre o grafo direcionado G=(N,E), é apresentado a seguir.…”

Section: Modelo Matemáticounclassified

Algoritmos Para O Problema Não Capacitado De Fluxos Com Custos Fixos Nos Arcos: Uma Comparação Estatística

Cruz¹,

Colosimo²,

Mateus

2001

Pesqui. Oper.

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ResumoEste trabalho tem como propósito a apresentação de resultados de uma comparação empírica entre algoritmos, sendo este um dos assuntos mais recorrentes na área de desenvolvimento de algoritmos. Os algoritmos sob estudo são para resolver um problema de otimização em redes, importante pelas suas aplicações potenciais em sistemas de telefonia e transporte, o problema não capacitado de fluxos com custos fixos nos arcos (NCFCF), uma generalização do clássico problema de Steiner em grafos. Para tal, são utilizadas ferramentas estatísticas conhecidas tais como planejamento de experimentos, análise de variância e intervalos de confiança, mas não comumente empregadas neste tipo de estudo. O problema NCFCF é apresentado em uma modelagem de programação matemática inteira mista, baseada na qual os algoritmos sob consideração são apresentados. Uma descrição do planejamento de experimentos adequado a este tipo de estudo é apresentada e é ilustrado o uso da técnica estatística baseado em cuja análise foi possível classificar os algoritmos sob consideração. Palavras-chave: otimização em redes, problemas de Steiner, ANOVA. AbstractThis paper is concerned about empirical comparisons of algorithms, one of the most recurrent issues in the field of design of algorithms. The problem under consideration is the uncapacitated fixed-charge network flow (UFCNF) problem, a generalization of the classic Steiner problem in graphs. The UFCNF is very important in the practical point of view because of its potential applications for telecommunication and transportation system design. In order to compare the algorithms for the UFCNF problem, we make use of well known and well established statistical tools namely the design of experiments, analysis of variance, and confidence intervals, but rarely applied in such studies. A mixed-integer mathematical programming formulation is presented for the UFCNF problem and the design of experiments suitable for the empirical study is detailed. Finally, the statistical analysis based on which the algorithms could be classified is presented.

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“…A variety of heuristics can be found in Stadtler (2003), Pochet and Van Vyve (2004), Federgruen et al (2007), and Akartunalı and Miller (2009). Mathematical programming approaches have mainly involved adding valid inequalities (e.g., Barany et al 1984;Constantino 1996;Wolsey 1988, 1994) and extended reformulations of the problem (e.g., Krarup and Bilde 1977, Eppen and Martin 1987, Rardin and Wolsey 1993, although few studies facilitate other techniques, such as Lagrangian relaxation (e.g., Billington et al 1986) and Dantzig-Wolfe decomposition (e.g., Bitran andMatsuo 1986, Degraeve andJans 2007). The interested reader is also referred to Belvaux and Wolsey (2001) for modeling and reformulation issues and to Pochet and Wolsey (2006) for an excellent, thorough review of lot-sizing problems and solution methods used.…”

Section: Introductionmentioning

confidence: 99%

Local Cuts and Two-Period Convex Hull Closures for Big-Bucket Lot-Sizing Problems

Akartunalı

Fragkos

Miller³

et al. 2016

INFORMS Journal on Computing

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D espite the significant attention they have drawn, big-bucket lot-sizing problems remain notoriously difficult to solve. Previous literature contained results (computational and theoretical) indicating that what makes these problems difficult are the embedded single-machine, single-level, multiperiod submodels. We therefore consider the simplest such submodel, a multi-item, two-period capacitated relaxation. We propose a methodology that can approximate the convex hulls of all such possible relaxations by generating violated valid inequalities. To generate such inequalities, we separate two-period projections of fractional linear programming solutions from the convex hulls of the two-period closure we study. The convex hull representation of the twoperiod closure is generated dynamically using column generation. Contrary to regular column generation, our method is an outer approximation and can therefore be used efficiently in a regular branch-and-bound procedure. We present computational results that illustrate how these two-period models could be effective in solving complicated problems.

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Valid inequalities and projecting the multicommodity extended formulation for uncapacitated fixed charge network flow problems

Cited by 65 publications

References 8 publications

A Polyhedral Study of Multiechelon Lot Sizing with Intermediate Demands

A Polyhedral Study of Multiechelon Lot Sizing with Intermediate Demands

Algoritmos Para O Problema Não Capacitado De Fluxos Com Custos Fixos Nos Arcos: Uma Comparação Estatística

Local Cuts and Two-Period Convex Hull Closures for Big-Bucket Lot-Sizing Problems

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