Fixed-charge Capacitated Network Design Problem Research Articles

Node-based Lagrangian relaxations for multicommodity capacitated fixed-charge network design

Classical Lagrangian relaxations for the multicommodity capacitated fixed-charge network design problem are the so-called flow and knapsack relaxations, where the resulting Lagrangian subproblems decompose by commodities and by arcs, respectively. We introduce node-based Lagrangian relaxations, where the resulting Lagrangian subproblem decomposes by nodes. We show that the Lagrangian dual bounds of these relaxations improve upon the linear programming relaxation bound, known to be equal to the Lagrangian dual bounds for the flow and knapsack relaxations. We also develop a Lagrangian matheuristic to compute upper bounds. The computational results on a set of benchmark instances show that the Lagrangian matheuristic is competitive with the state-of-the-art heuristics from the literature.

A Learning-Based Matheuristic for Stochastic Multicommodity Network Design

This paper proposes a solution approach for the multicommodity capacitated fixed-charge network design problem with uncertain demand modeled as a two-stage stochastic program. The proposed learning-based matheuristic combines heuristic search techniques with mathematical programming. It provides a systematic approach to identifying structures of good-quality solutions by gradually considering scenarios and their influences on design decisions. Extensive computational experiments illustrate the efficiency of the proposed matheuristic in obtaining high-quality solutions with limited computational efforts.

MIP Neighborhood Search Heuristics for a Capacitated Fixed-Charge Network Design Problem

The fixed-charge capacitated multicommodity network design problem is a fundamental optimization problem arising in many network configurations. The solution of the problem provides an appropriate network design as well as routes of multicommodity flows aimed at minimizing the total cost, which is the sum of the flow costs and fixed-charge costs over a network with limited arc capacities. In the present paper, we introduce a combined approach with a capacity scaling procedure for finding an initial feasible solution and an MIP neighborhood search for improving the solutions. Besides, we modify the procedure for application to large-scale problems. Computational experiments demonstrate the effectiveness of the proposed approach, and high-quality solutions are obtained for two problem sets from the literature.

Matheuristics based on iterative linear programming and slope scaling for multicommodity capacitated fixed charge network design

We present new matheuristics for the multicommodity capacitated fixed-charge network design problem (MCND). The matheuristics are based on combining iterative linear programming (ILP) methods and slope scaling (SS) heuristics. Each iteration alternates between solving a linear program obtained by adding pseudo-cuts and a restricted mixed-integer programming (MIP) model. The SS heuristic is used as a warm start to a state-of-the-art generic method that solves the restricted MIP model. The resulting ILP/SS matheuristics are compared against state-of-the-art heuristics for the MCND on a set of large-scale difficult instances. The computational results show that the approach is competitive: when performed for a time limit of 1 hour, it finds more best solutions than any other heuristic, using comparable running times; when performed for a time limit of 5 hours, it identifies an optimal solution for each instance for which an optimal solution is known and it is able to find new best solutions for some very hard instances.

Commodity Representations and Cut-Set-Based Inequalities for Multicommodity Capacitated Fixed-Charge Network Design

We improve the mixed-integer programming formulation of the multicommodity capacitated fixed-charge network design problem by incorporating valid inequalities into a cutting-plane algorithm. We use five classes of known valid inequalities: the strong, cover, minimum cardinality, flow cover, and flow pack inequalities. The first class is particularly useful when a disaggregated representation of the commodities is chosen, and the last four are expressed in terms of network cut sets. We develop efficient separation and lifting procedures for these classes of inequalities. We present computational results on a large set of instances of various characteristics, allowing us to measure the impact of the different classes of valid inequalities on the quality of the lower bounds, in particular with respect to the representation of the commodities.

Formulations for the nonbifurcated hop-constrained multicommodity capacitated fixed-charge network design problem

This paper addresses the multicommodity capacitated fixed-charge network design problem with nonbifurcated flows and hop constraints. We present and compare mathematical programming formulations for this problem and we study different relaxations: Lagrangean relaxations, linear programming relaxations, and partial relaxations of the integrality constraints. In particular, we show that the Lagrangean bound obtained by relaxing the flow conservation equations is tighter than the linear programming relaxation bound. We present computational results on a large set of randomly generated instances.

Branch-and-price-and-cut for large-scale multicommodity capacitated fixed-charge network design

We present a branch-and-price-and-cut algorithm for solving large-scale instances of the multicommodity capacitated fixed-charge network design problem. We assume good feasible solutions are already known and we focus on an efficient algorithm for proving the optimality of the solutions. The restricted master problem solved at each column generation iteration is obtained directly from the compact arc-based model by considering only a subset of the commodity flow variables. The pricing subproblem corresponds to a Lagrangian relaxation of the flow conservation and capacity constraints, leaving in the Lagrangian subproblem only the strong inequalities. The column generation procedure is completed by a cut generation step based on strong inequalities. The resulting column-and-row generation procedure is embedded within an enumerative scheme. Computational experiments on a large set of randomly generated instances are presented and analyzed.

A two-phase heuristic algorithm for the fixed-charge capacitated network design problem with turn penalties

In this paper, we consider the fixed-charge capacitated network design problem with turn penalties. The objective of the problem is to minimize the sum of flow costs for routing the commodities demand, fixed costs for using arcs and penalty costs for flows with 90-degree turns. A mixed integer programming model is presented for the problem and a two-phase heuristic algorithm is suggested to solve the problem. The arc-labeling algorithm is used to find a flow path with a minimum cost and it is embedded in the suggested heuristic algorithm as a sub-routine. In the suggested algorithm, an initial flow path network is obtained by a construction method and it is improved by an iterative improvement method. To evaluate the performance of the suggested algorithm, computational experiments are performed on randomly generated test problems. Results of computational experiments show that the suggested algorithm finds near-optimal solutions in a short computation time even for large-sized problems.

A local branching heuristic for the capacitated fixed-charge network design problem

We describe in this article a new heuristic approach for the capacitated fixed-charge network design problem. This is a well known NP -hard problem that has been extensively studied. We propose to use local branching to tackle it. Local branching is a novel metaheuristic technique presented by Fischetti and Lodi [Local branching. Mathematical Programming Series B 2003;98:23–47] that utilizes a general MIP solver to explore neighborhoods. Our implementation is tested on a large set of benchmark instances. The results show that the method outperforms the best heuristic approaches found in the literature.

A Slope Scaling/Lagrangean Perturbation Heuristic with Long-Term Memory for Multicommodity Capacitated Fixed-Charge Network Design

This paper describes a slope scaling heuristic for solving the multicomodity capacitated fixed-charge network design problem. The heuristic integrates a Lagrangean perturbation scheme and intensification/diversification mechanisms based on a long-term memory. Although the impact of the Lagrangean perturbation mechanism on the performance of the method is minor, the intensification/diversification components of the algorithm are essential for the approach to achieve good performance. The computational results on a large set of randomly generated instances from the literature show that the proposed method is competitive with the best known heuristic approaches for the problem. Moreover, it generally provides better solutions on larger, more difficult, instances.

Revue Des Inégalités Valides Pertinentes Aux Problèmes Des Conception De Réseaux

The objective of this paper is to present a survey of relevant valid inequalities for the multi-commodity capacitated fixed-charge network design problem. We present first a review of the relevant literature on relaxations and heuristic approaches. We observe that these approaches alone are not sufficient to solve large-scale instances of the problem. However, a combination or these approaches and polyhedral methods appears very promising. Then, we present three classes of relevant valid inequalities for the problem that have been studied for related problems. We present as well experimental results on the general implementations of some of these inequalities in a state-of-the-art commercial software. These experiments point towards the necessity of profound polyhedral studies in order to solve efficiently large-scale instances of the problem.

A note on “a dual-ascent approach to the fixed-charge capacitated network design problem”

We show by a counterexample that the dual-ascent procedure proposed by Herrmann, Ioannou, Minis and Proth in a 1996 issue of the European Journal of Operational Research is incorrect in the sense that it does not generate a valid lower bound to the optimal value of fixed-charge capacitated network design problems. We provide a correct dual-ascent procedure based on the same ideas and we give an interpretation of it in terms of a simple Lagrangean relaxation. Although correct, this procedure is not effective, as in general, it provides a less tighter bound than the linear programming relaxation.

Bundle-based relaxation methods for multicommodity capacitated fixed charge network design

To efficiently derive bounds for large-scale instances of the capacitated fixed-charge network design problem, Lagrangian relaxations appear promising. This paper presents the results of comprehensive experiments aimed at calibrating and comparing bundle and subgradient methods applied to the optimization of Lagrangian duals arising from two Lagrangian relaxations. This study substantiates the fact that bundle methods appear superior to subgradient approches because they converge faster and are more robust relative to different relaxations, problem characteristics, and selection of the initial parameter values. It also demonstrates that effective lower bounds may be computed efficiently for large-scale instances of the capacitated fixed-charge network design problem. Indeed, in a fraction of the time required by a standard simplex approach to solve the linear programming relaxation, the methods we present attain very high-quality solutions.

Benders-and-cut algorithm for fixed-charge capacitated network design problem

We develop a Benders-and-cut algorithm for fixed-charge capacitated network design problem which incorporates both Benders cuts and polyhedral cuts into an implicit branch-and-bound algorithm. The algorithm is tested over a wide range of problem instances with varying sizes and traffic conditions. It is shown that this new algorithm’s performance is competitive compared with existing algorithms, and that Benders cuts are more effective under heavy traffic load, while cut set inequalities are more effective under light traffic load.

A dual ascent approach to the fixed-charge capacitated network design problem

In this paper we consider the problem of constructing a network over which a number of commodities are to be transported. Fixed costs are associated to the construction of network arcs and variable costs are associated to routing of commodities. In addition, one capacity constraint is related to each arc. The problem is to determine a network design that minimizes the total cost; i.e., it balances the construction and operating costs. A dual ascent procedure for finding improved lower bounds and near-optimal solutions for the fixed-charge capacitated network design problem is proposed. The method is shown to generate tighter lower bounds than the linear programming relaxation of the problem.