Fixed-charge Network Design Problem Research Articles

New partial aggregations for multicommodity network flow problems: An application to the fixed-charge network design problem

When solving hard multicommodity network flow problems using an LP-based approach, the number of commodities is a driving factor in the speed at which the LP can be solved, as it is linear in the number of constraints and variables. The conventional approach to improve the solve time of the LP relaxation of a Mixed Integer Programming (MIP) model that encodes such an instance is to aggregate all commodities that have the same origin or the same destination. However, the bound of the resulting LP relaxation can significantly worsen, which tempers the efficiency of aggregating techniques. In this paper, we introduce the concept of partial aggregation of commodities that aggregates commodities over a subset of the network instead of the conventional aggregation over the entire underlying network. This offers a high level of control on the trade-off between size of the aggregated MIP model and quality of its LP bound. We apply the concept of partial aggregation to two different MIP models for the multicommodity network design problem. Our computational study on benchmark instances confirms that the trade-off between solve time and LP bound can be controlled by the level of aggregation, and that choosing a good trade-off can allow us to solve the original large-scale problems faster than without aggregation or with full aggregation.

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.

Deep Dual-Optimal Inequalities for Generalized Capacitated Fixed-Charge Network Design Problems

Capacitated fixed-charge network design problems and generalizations, such as service network design problems, have a wide range of applications but are known to be very difficult to solve. Many exact and heuristic algorithms to solve these problems rely on column-and-row generation (CRG), which frequently suffer from primal degeneracy. We present a set of dual inequalities, equivalent to a simple primal relaxation, that speed up CRG algorithms for generalized capacitated fixed charge network design problems. We investigate the impact of the dual inequalities theoretically as well as experimentally. For practical applications, the presented technique is simple to implement, has no additional computational cost and can accelerate CRG by orders of magnitude, depending on the problem size and structure.

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.

Exact algorithms based on Benders decomposition for multicommodity uncapacitated fixed-charge network design

This paper presents two exact algorithms based on Benders decomposition for solving the multicommodity uncapacitated fixed-charge network design problem. The first is a branch-and-cut algorithm based on a Benders reformulation enhanced with an in-tree matheuristic to obtain improved feasible solutions, valid inequalities in the projected master problem space to close the linear programming gap, and a tailored core point selection criterion from which significant time savings are obtained. The second exact algorithm exploits the problem’s structure to combine a cut-and-solve strategy with Benders decomposition. Extensive computational experiments show both exact algorithms provide a speed-up of up to three orders of magnitude compared to a state-of-the-art general-purpose MIP solver’s branch-and-cut and blackbox Benders decomposition algorithms.

On two new formulations for the fixed charge network design problem with shortest path constraints

We study the fixed charge network design problem with shortest path constraints which is modeled as a bi-level program. We first review three one-level formulations obtained by applying the complementarity slackness theorem, Bellman’s optimality conditions and cycle elimination constraints. We propose two new binary integer programming (BILP) formulations inspired by path and cycle inequalities. The two formulations have exponential numbers of constraints. We incorporate the path and the cycle based formulations in a branch-and-cut algorithm and in another cutting-plane based method. Numerical experiments are performed on real instances, and random data sets generated with different criteria to examine the difficulty of the instances. The results show that the proposed cutting plane algorithms can solve up to 19% more instances than the classic branch-and-bound algorithms.

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.

The impact of filtering in a branch-and-cut algorithm for multicommodity capacitated fixed charge network design

In this paper, we present a state-of-the-art branch-and-cut (B&C) algorithm for the multicommodity capacitated fixed charge network design problem (MCND). This algorithm combines bounding and branching procedures inspired by the latest developments in mixed-integer programming (MIP) software tools. Several filtering methods that exploit the structure of the MCND are also developed and embedded within the B&C algorithm. These filtering methods apply inference techniques to forbid combinations of values for some variables. This can take the form of adding cuts, reducing the domains of the variables, or fixing the values of the variables. Our experiments on a large set of randomly generated instances show that an appropriate selection of filtering techniques allows the B&C algorithm to perform better than the variant of the algorithm without filtering. These experiments also show that the B&C algorithm, with or without filtering, is competitive with a state-of-the-art MIP solver.

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.

An Efficient Matheuristic for the Multicommodity Fixed-Charge Network Design Problem

In this paper we study the Multicommodity Fixed-Charge Network Design problem. We propose an Iterative linear programming-based heuristic for solving this NP hard problem. The proposed heuristics have been tested on the benchmark instances from the literature. The quality of solutions obtained by each of them has been disclosed comparing them with corresponding solutions of the current state-of-the-art heuristics, i.e., Cycle-Based Evolutionary algorithms.

A Rolling Horizon Heuristic for the Multiperiod Network Design and Routing Problem

The capacitated fixed‐charge network design (FCND) problem considers the simultaneous optimization of capacity installation and routing of traffic where a fixed cost is paid for opening a link and a linear routing cost is paid for sending traffic flow(s) on that link. The routing decisions must be performed such that the traffic flows remain bounded by the installed link capacities. The FCND problem appears as a particular case of the combined multiperiod network design and traffic flow routing problem with time‐dependent demands formulated in this article as a mixed integer linear program (MILP). A compact formulation based on the aggregation of traffic flows per destination and an extended formulation, where flows are decomposed by origin‐destination pairs while keeping the requirement of destination‐based routing, have been proposed in D. Papadimitriou and B. Fortz, IEEE International Conference on Communications (2014), pp.1124–1130 and IEEE Global Communications Conference (2014), pp.1303–1309, respectively. In this article, we propose to resolve this computationally challenging problem by means of the rolling horizon heuristic with the objective to decrease the computational time while degrading as less as possible the quality of the solution. The resulting improvements enable to progressively overcome the computational limits encountered when solving such problem, in particular, when the network size and number of periods increase. The improvements provided by the rolling horizon heuristic can be further exploited by extending the proposed model to account for different patterns of failures that may affect installed arcs over time. For this purpose, our generalized MILP formulation comprises a time‐variable link maintenance cost function. We further analyze the quality of the results for the proposed formulation with different link maintenance cost functions with the objective to derive the best arc replacement strategy. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(4), 364–379 2015

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.

Benders decomposition algorithms for the fixed-charge relay network design in telecommunications

In translucent networks, signal regenerators that operate based on optical/electrical/optical (O/E/O) conversion are employed to handle the problems of physical impairments and resulting signal quality issues in long-distance transmissions. Regenerators are placed at the relay nodes in a translucent network in such a way that the optical signals are not transmitted farther than a threshold distance without a 3R regeneration (reamplification-reshaping-retiming). In this study, we consider a fixed-charge relay network design (FCRND) problem whose applications are found mainly in long distance translucent optical telecommunication networks. From a modeling perspective, FCRND combines prominent features of the uncapacitated single-assignment hub location (location of relays points) and the fixed-charge network design problems (choice of transmission links in the network and routing of signals) while additionally considering distance-based coverage constraints. We devise Benders decomposition based solution methodologies in which the algorithmic performance is further enhanced by devising strengthened and disaggregated Benders cuts, surrogate constraints for the master problem, and an upper bound heuristic to both obtain and tighten optimality Benders cuts. We obtain solutions within 2.0 % of optimality in very reasonable times as we illustrate in our computational study.

Accelerating benders decomposition with heuristicmaster problem solutions

In this paper, a general scheme for generating extra cuts during the execution of a Benders decomposition algorithm is presented. These cuts are based on feasible and infeasible master problem solutions generated by means of a heuristic. This article includes general guidelines and a case study with a fixed charge network design problem. Computational tests with instances of this problem show the efficiency of the strategy. The most important aspect of the proposed ideas is their generality, which allows them to be used in virtually any Benders decomposition implementation.

New models of the generalized fixed-charge network design problem

We consider in this paper the generalized fixed-charge network design (GFCND) problem in which we are interested to find the cheapest backbone network connecting exactly one hub from each of the given clusters. The GFCND problem belongs to the class of generalized combinatorial optimization problems. We describe two mixed integer programming formulations of the GFCND problem. Based on one of the new proposed formulations, we solve the GFCND problem to optimality using CPLEX for problems with up to 30 clusters and 200 nodes. Computational results are reported and compared with those from the literature.

An interior-point Benders based branch-and-cut algorithm for mixed integer programs

We present an interior-point branch-and-cut algorithm for structured integer programs based on Benders decomposition and the analytic center cutting plane method (ACCPM). We show that the ACCPM based Benders cuts are both pareto-optimal and valid for any node of the branch-and-bound tree. The valid cuts are added to a pool of cuts that is used to warm-start the solution of the nodes after branching. The algorithm is tested on two classes of problems: the capacitated facility location problem and the multicommodity capacitated fixed charge network design problem. For the capacitated facility location problem, the proposed approach was on average 2.5 times faster than Benders-branch-and-cut and 11 times faster than classical Benders decomposition. For the multicommodity capacitated fixed charge network design problem, the proposed approach was 4 times faster than Benders-branch-and-cut while classical Benders decomposition failed to solve the majority of the tested instances.

Approximability of unsplittable shortest path routing problems

AbstractIn this article, we discuss the relation of unsplittable shortest path routing (USPR) to other routing schemes and study the approximability of three USPR network planning problems. Given a digraph D = (V,A) and a set K of directed commodities, an USPR is a set of flow paths P, (s,t) ∈ K, such that there exists a metric λ = (λa) ∈ Z with respect to which each P is the unique shortest (s,t)‐path. In the Min‐Con‐USPR problem, we seek an USPR that minimizes the maximum congestion over all arcs. We show that this problem is NP‐hard to approximate within a factor of O(|V|1−ϵ), but polynomially approximable within min(|A|,|K|) in general and within O(1) if the underlying graph is an undirected cycle or a bidirected ring. We also construct examples where the minimum congestion that can be obtained by USPR is a factor of Ω(|V|2) larger than that achievable by unsplittable flow routing or by shortest multipath routing, and a factor of Ω(|V|) larger than that achievable by unsplittable source‐invariant routing. In the CAP‐USPR problem, we seek a minimum cost installation of integer arc capacities that admit an USPR of the given commodities. We prove that this problem is NP‐hard to approximate within 2 − ϵ even in the undirected case, and we devise approximation algorithms for various special cases. The fixed charge network design problem FC‐USPR, where the task is to find a minimum cost subgraph of D whose fixed arc capacities admit an USPR of the commodities, is shown to be NPO‐complete. All three problems are of great practical interest in the planning of telecommunication networks that are based on shortest path routing protocols. Our results indicate that they are harder than the corresponding unsplittable flow or shortest multi‐path routing problems. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009

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.