The Robust Network Loading Problem under Hose Demand Uncertainty: Formulation, Polyhedral Analysis, and Computations

Cataloged from PDF version of article.We consider the network loading problem (NLP) under a polyhedral uncertainty description of traffic demands. After giving a compact multicommodity flow formulation of the problem, we state a decomposition property obtained from projecting out the flow variables. This property considerably simplifies the resulting polyhedral analysis and computations by doing away with metric inequalities. Then we focus on a specific choice of the uncertainty description, called the “hose model,” which specifies aggregate traffic upper bounds for selected endpoints of the network. We study the polyhedral aspects of the NLP under hose demand uncertainty and use the results as the basis of an efficient branch-and-cut algorithm. The results of extensive computational experiments on well-known network design instances are reported

Decomposition Methods for Network Design

AbstractNetwork design applications are prevalent in transportation and logistics. We consider the multicommodity capacitated fixed-charge network design problem (MCND), a generic model that captures three important features of network design applications: the interplay between investment and operational costs, the multicommodity aspect, and the presence of capacity constraints. We focus on mathematical programming approaches for the MCND and present three classes of methods that have been used to solve large-scale instances of the MCND: a cutting-plane method, a Benders decomposition algorithm, and Lagrangian relaxation approaches

Lagrangian-based methods for single and multi-layer multicommodity capacitated network design

Le problème de conception de réseau avec coûts fixes et capacités (MCFND) et le problème de conception de réseau multicouches (MLND) sont parmi les problèmes de conception de réseau les plus importants. Dans le problème MCFND monocouche, plusieurs produits doivent être acheminés entre des paires origine-destination différentes d’un réseau potentiel donné. Des liaisons doivent être ouvertes pour acheminer les produits, chaque liaison ayant une capacité donnée. Le problème est de trouver la conception du réseau à coût minimum de sorte que les demandes soient satisfaites et que les capacités soient respectées. Dans le problème MLND, il existe plusieurs réseaux potentiels, chacun correspondant à une couche donnée. Dans chaque couche, les demandes pour un ensemble de produits doivent être satisfaites. Pour ouvrir un lien dans une couche particulière, une chaîne de liens de support dans une autre couche doit être ouverte. Nous abordons le problème de conception de réseau multiproduits multicouches à flot unique avec coûts fixes et capacités (MSMCFND), où les produits doivent être acheminés uniquement dans l’une des couches. Les algorithmes basés sur la relaxation lagrangienne sont l’une des méthodes de résolution les plus efficaces pour résoudre les problèmes de conception de réseau. Nous présentons de nouvelles relaxations à base de noeuds, où le sous-problème résultant se décompose par noeud. Nous montrons que la décomposition lagrangienne améliore significativement les limites des relaxations traditionnelles. Les problèmes de conception du réseau ont été étudiés dans la littérature. Cependant, ces dernières années, des applications intéressantes des problèmes MLND sont apparues, qui ne sont pas couvertes dans ces études. Nous présentons un examen des problèmes de MLND et proposons une formulation générale pour le MLND. Nous proposons également une formulation générale et une méthodologie de relaxation lagrangienne efficace pour le problème MMCFND. La méthode est compétitive avec un logiciel commercial de programmation en nombres entiers, et donne généralement de meilleurs résultats.The multicommodity capacitated fixed-charge network design problem (MCFND) and the multilayer network design problem (MLND) are among the most important network design problems. In the single-layer MCFND problem, several commodities have to be routed between different origin-destination pairs of a given potential network. Appropriate capacitated links have to be opened to route the commodities. The problem is to find the minimum cost design and routing such that the demands are satisfied and the capacities are respected. In the MLND, there are several potential networks, each at a given layer. In each network, the flow requirements for a set of commodities must be satisfied. However, the selection of the links is interdependent. To open a link in a particular layer, a chain of supporting links in another layer has to be opened. We address the multilayer single flow-type multicommodity capacitated fixed-charge network design problem (MSMCFND), where commodities are routed only in one of the layers. Lagrangian-based algorithms are one of the most effective solution methods to solve network design problems. The traditional Lagrangian relaxations for the MCFND problem are the flow and knapsack relaxations, where the resulting Lagrangian subproblems decompose by commodity and by arc, respectively. We present new node-based relaxations, where the resulting subproblem decomposes by node. We show that the Lagrangian dual bound improves significantly upon the bounds of the traditional relaxations. We also propose a Lagrangian-based algorithm to obtain upper bounds. Network design problems have been the object of extensive literature reviews. However, in recent years, interesting applications of multilayer problems have appeared that are not covered in these surveys. We present a review of multilayer problems and propose a general formulation for the MLND. We also propose a general formulation and an efficient Lagrangian-based solution methodology for the MMCFND problem. The method is competitive with (and often significantly better than) a state-of-the-art mixedinteger programming solver on a large set of randomly generated instances

Modeling and Solving the Capacitated Network Loading Problem

This paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry. In this problem, given point-topoint demand between various pairs of nodes of a network must be met by installing (loading) capacitated facilities on the arcs. The facilities are chosen from a small set of alternatives and loading a particular facility incurs an arc specific and facility dependent cost. The problem is to determine the configuration of facilities to be loaded on the arcs of the network that will satisfy the given demand at minimum cost. Since we need to install (load) facilities to carry the required traffic, we refer to the problem as the network loading problem. In this paper, we develop modeling and solution approaches for the problem. We consider two approaches for solving the underlying mixed integer programming model: (i) a Lagrangian relaxation strategy, and (ii) a cutting plane approach that uses three classes of valid inequalities that we identify for the problem. In particular, we show that a linear programming formulation that includes the valid inequalities always approximates the value of the mixed integer program at least as well as the Lagrangian relaxation bound (as measured by the gaps in the objective functions). We also examine the computational effectiveness of these inequalities on a set of prototypical telecommunications data. The computational results show that the addition of these inequalities considerably improves the gap between the integer programming formulation of the problem and its linear programming relaxation: for 6 - 15 node problems from an average of 25% to an average of 8%. These results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

Multicommodity capacitated network design

Network design models have wide applications in telecommunications and transportation planning; see, for example, the survey articles by Magnanti and Wong (1984), Minoux (1989), Chapter 16 of the book by Ahuja, Magnanti and Orlin (1993), Section 13 of Ahuja et al. (1995). In particular, Gavish (1991) and Balakrishnan et al. (1991) present reviews of important applications in telecommunications. In many of these applications, it is required to send flows (which may be fractional) to satisfy demands given arcs with existing capacities, or to install, in discrete amounts, additional facilities with fixed capacities. In doing so, one pays a price not only for routing flows, but also for using an arc or installing additional facilities. The objective is then to determine the optimal amounts of flows to be routed and the facilities to be installed. Document type: Part of book or chapter of boo

Modeling the Multicommodity Multimodal Routing Problem with Schedule-Based Services and Carbon Dioxide Emission Costs

We explore a freight routing problem wherein the aim is to assign optimal routes to move commodities through a multimodal transportation network. This problem belongs to the operational level of service network planning. The following formulation characteristics will be comprehensively considered: (1) multicommodity flow routing; (2) a capacitated multimodal transportation network with schedule-based rail services and time-flexible road services; (3) carbon dioxide emissions consideration; and (4) a generalized costs optimum oriented to customer demands. The specific planning of freight routing is thus defined as a capacitated time-sensitive multicommodity multimodal generalized shortest path problem. To solve this problem systematically, we first establish a node-arc-based mixed integer nonlinear programming model that combines the above formulation characteristics in a comprehensive manner. Then, we develop a linearization method to transform the proposed model into a linear one. Finally, a computational experiment from the Chinese inland container export business is presented to demonstrate the feasibility of the model and linearization method. The computational results indicate that implementing the proposed model and linearization method in the mathematical programming software Lingo can effectively solve the large-scale practical multicommodity multimodal transportation routing problem

Shortest Paths, Network Design and Associated Polyhedra

We study a specialized version of network design problems that arise in telecommunication, transportation and other industries. The problem, a generalization of the shortest path problem, is defined on an undirected network consisting of a set of arcs on which we can install (load), at a cost, a choice of up to three types of capacitated facilities. Our objective is to determine the configuration of facilities to load on each arc that will satisfy the demand of a single commodity at the lowest possible cost. Our results (i) demonstrate that the single-facility loading problem and certain "common breakeven point" versions of the two-facility and three-facility loading problems are polynomially solvable as a shortest path problem; (ii) show that versions of the twofacility loading problem are strongly NP-hard, but that a shortest path solution provides an asymptotically "good" heuristic; and (iii) characterize the optimal solution (that is, specify a linear programming formulation with integer solutions) of the common breakeven point versions of the two-facility and three-facility loading problems. In this development, we introduce two new families of facets, give geometric interpretations of our results, and demonstrate the usefulness of partitioning the space of the problem parameters to establish polyhedral integrality properties. Generalizations of our results apply to (i) multicommodity applications and (ii) situations with more than three facilities

Design of a Distribution Network Using Primal-Dual Decomposition

Amethodtosolvethedesignofadistributionnetworkforbottleddrinkscompanyisintroduced.Thedistributionnetworkproposed includes three stages: manufacturing centers, consolidation centers using cross-docking, and distribution centers. The problem is formulated using a mixed-integer programming model in the deterministic and single period contexts. Because the problem considersseveralelementsineachstage,adirectsolutionisverycomplicated.Formedium-to-largeinstancestheproblemfallsinto large scale. Based on that, a primal-dual decomposition known as cross decomposition is proposed in this paper. This approach allows exploring simultaneously the primal and dual subproblems of the original problem. A comparison of the direct solution withamixed-integerlinealprogrammingsolverversusthecrossdecompositionisshownforseveralrandomlygeneratedinstances. Resultsshowthegoodperformanceofthemethodproposed