Multicommodity Capacitated Network Design Problem Research Articles

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.

A combined fast greedy heuristic for the capacitated multicommodity network design problem

The capacitated multicommodity network design problem represents a network design system and has a wide range of real-life applications, such as the construction of logistics networks, transportation networks, communication networks, and production networks. In this article, we introduce a fast greedy algorithm for solving the capacitated multicommodity network design problem. The greedy algorithm is based on link-rerouting and partial link-rerouting heuristics for the uncapacitated multicommodity network design problem. This algorithm involves a capacity scaling for reducing the number of candidate arcs and a restricted branch-and-bound for improving solutions. The algorithm succeeds in finding good solutions within a short computation time. The average computation time for solving benchmark problem instances is only several tens of seconds.

Revisiting Lagrangian relaxation for network design

We consider a general network design model for which we compare theoretically different Lagrangian relaxations. Fairly general assumptions on the model are proposed, allowing us to generalize results obtained for special cases. The concepts are illustrated on two classical network design problems, the fixed-charge multicommodity capacitated network design problem and the single-source capacitated facility location problem. We also show that, even though our assumptions might not be verified, it is often possible to derive extended reformulations that satisfy them. We study the impact of such extended reformulations on Lagrangian relaxations, by focusing on two particular cases, the hop-constrained fixed-charge multicommodity capacitated network design problem and the multicommodity capacitated network design problem.

Accelerating the Benders Decomposition Method: Application to Stochastic Network Design Problems

This paper describes a Benders decomposition algorithm capable of efficiently solving large-scale instances of the well-known multicommodity capacitated network design problem with demand uncertainty. The problem is important because it models many applications, including telecommunications, transportation, and logistics. This problem has been tackled in the literature with meta-heuristics and exact methods, but many benchmark instances, even though of moderate size, remain unsolved. To successfully apply the Benders method to these instances, we propose various acceleration techniques, including the use of cutting planes, partial decomposition, heuristics, stronger cuts, reduction and warm-start strategies. Extensive computational experiments on benchmark instances were conducted to evaluate the efficiency and robustness of the algorithm as well as of the proposed strategies. The numerical results confirm the superiority of the proposed algorithm over existing ones.

On the computational efficiency of subgradient methods: a case study with Lagrangian bounds

Subgradient methods (SM) have long been the preferred way to solve the large-scale Nondifferentiable Optimization problems arising from the solution of Lagrangian Duals (LD) of Integer Programs (IP). Although other methods can have better convergence rate in practice, SM have certain advantages that may make them competitive under the right conditions. Furthermore, SM have significantly progressed in recent years, and new versions have been proposed with better theoretical and practical performances in some applications. We computationally evaluate a large class of SM in order to assess if these improvements carry over to the IP setting. For this we build a unified scheme that covers many of the SM proposed in the literature, comprised some often overlooked features like projection and dynamic generation of variables. We fine-tune the many algorithmic parameters of the resulting large class of SM, and we test them on two different LDs of the Fixed-Charge Multicommodity Capacitated Network Design problem, in order to assess the impact of the characteristics of the problem on the optimal algorithmic choices. Our results show that, if extensive tuning is performed, SM can be competitive with more sophisticated approaches when the tolerance required for solution is not too tight, which is the case when solving LDs of IPs.

A cycle-based evolutionary algorithm for the fixed-charge capacitated multi-commodity network design problem

This paper presents an evolutionary algorithm for the fixed-charge multicommodity network design problem (MCNDP), which concerns routing multiple commodities from origins to destinations by designing a network through selecting arcs, with an objective of minimizing the fixed costs of the selected arcs plus the variable costs of the flows on each arc. The proposed algorithm evolves a pool of solutions using principles of scatter search, interlinked with an iterated local search as an improvement method. New cycle-based neighborhood operators are presented which enable complete or partial re-routing of multiple commodities. An efficient perturbation strategy, inspired by ejection chains, is introduced to perform local compound cycle-based moves to explore different parts of the solution space. The algorithm also allows infeasible solutions violating arc capacities while performing the “ejection cycles”, and subsequently restores feasibility by systematically applying correction moves. Computational experiments on benchmark MCNDP instances show that the proposed solution method consistently produces high-quality solutions in reasonable computational times.

English

The capacitated multi-commodity network design problem (CMND) represents a generic network design model for applications used in designing construction and improvements to telecommunication, logistics, transportation, distribution, and production networks. This paper presents an approach combining capacity scaling and local branching for CMND. Capacity scaling is an approximate iterative solution approach for capacitated network problems based on the principle of changing arc capacities that depend on flow volumes on arcs. Local branching is a method of solving new restricted problems based on an exploration of solution neighborhoods defined as local branching constraints. Capacity scaling with a strong path flow based formulation including forcing constraints can produce high-quality solutions within a short period of time allotted for computation. By combined capacity scaling and local branching, the combined approach can offer one of the best current solutions compared to previous heuristics for CMND.

A Genetic Algorithm Based on Relaxation Induced Neighborhood Search in a Local Branching Framework for Capacitated Multicommodity Network Design

The fixed-charge Capacitated Multicommodity Network Design (CMND) is a well-known problem of both practical and theoretical significance. This article proposes the Genetic Algorithm (GA) cooperative Relaxation Induced Neighborhood Search (RINS) in a Local Branching (LB) framework for CMND problem. GA algorithm is started by initial population which is made by two parents obtain from hybrid LB and RINS algorithms. The basic idea of the proposed solution method is to use the GA algorithm to explore the search space and the hybrid LB and RINS methods to move from current solution to neighbor solution. Adapting the metaheuristic algorithm with RINS method to fit within an LB framework represents an interesting challenge. To evaluate the proposed algorithm, the standard problems with different sizes are used. The parameters of the algorithm are tuned by design of experiments. In order to prove the efficiency and effectiveness of the proposed algorithm, the results are compared with the best results available in the literature. The statistical analysis shows high performance of the proposed algorithm.

A hybrid solution method for fuzzy train formation planning

The train formation plan (TFP) determines the routing and frequency of trains, and assigns the demands to trains. In this paper, in order to consider the real-life condition of railways, a mathematical model with fuzzy costs is proposed for train formation planning in Iranian railway. In this fuzzy model, the costs are considered in three scenarios, namely optimistic, normal and pessimistic. The model is formulated based on the fixed-charge capacitated multicommodity network design problem. Since the TFP problem is NP-hard, an efficient hybrid algorithm combining local branching and relaxation induced neighborhood search methods is presented. A three-step method is applied for parameter tuning using design of experiments approach. To evaluate the efficiency and effectiveness of the proposed algorithm, the results are compared with those of the state-of-the-art optimization software.

A Cutting-Plane Neighborhood Structure for Fixed-Charge Capacitated Multicommodity Network Design Problem

In this paper, a cutting-plane neighborhood structure is proposed for the fixed-charge capacitated multicommodity network design (CMND) problem. In the proposed structure, different strategies are used to select an open arc in the incumbent solution to be closed. Then a linear programming (LP) model is generated on the basis of the modified incumbent solution by relaxing binary variables and adding new constraints. The generated LP solution is improved using different cutting-plane inequalities. Subsequently, a new sub-mixed integer programming (MIP) model is created by fixing a number of variables in the generated LP solution. Then the local branching algorithm is used to solve the sub-MIP model and its solution is considered as a neighboring solution. A tabu search algorithm is used to evaluate the proposed neighborhood structure. To tune the parameters of the tabu search algorithm, we have used the design of experiments method. Standard problems with different sizes are employed to evaluate the proposed tabu search algorithm. Results show the efficiency and effectiveness of the tabu search algorithm compared to the best methods found in the literature.

Cutting-Plane Matheuristic for Service Network Design with Design-Balanced Requirements

The paper introduces a cutting-plane matheuristic for the design-balanced capacitated multicommodity network design problem, one of the premier formulations for the service network design problem with asset management concerns increasingly faced by carriers within their tactical planning processes. The matheuristic combines a cutting-plane procedure efficiently computing tight lower bounds and a variable-fixing procedure feeding a MIP solver. Learning mechanisms embedded into the cutting-plane procedure provide the means to identify promising variables and thus both reduce the dimension of the problem instance, making it addressable by a MIP solver, and guide the latter toward promising solution spaces. Extensive computational experiments show the efficiency of the proposed procedures in obtaining high-quality solutions, outperforming the current best methods from the literature.

A stabilized structured Dantzig–Wolfe decomposition method

We discuss an algorithmic scheme, which we call the stabilized structured Dantzig–Wolfe decomposition method, for solving large-scale structured linear programs. It can be applied when the subproblem of the standard Dantzig–Wolfe approach admits an alternative master model amenable to column generation, other than the standard one in which there is a variable for each of the extreme points and extreme rays of the corresponding polyhedron. Stabilization is achieved by the same techniques developed for the standard Dantzig–Wolfe approach and it is equally useful to improve the performance, as shown by computational results obtained on an application to the multicommodity capacitated network design problem.

Solving Multicommodity Capacitated Network Design Problems Using Multiobjective Evolutionary Algorithms

Evolutionary algorithms can be applied to a variety of constrained network communication problems with centric type models. This paper shows that with real-world complex network communication problems of this type, sophisticated statistical search is required. This situation occurs due to the fact that these optimization problems are at least NP-complete. In order to appreciate the formal modeling of realistic communication networks, historical network design problems are presented and evolved into more complex real-world models with associated deterministic and stochastic solution approaches discussed. This discussion leads into the design of an innovative multiobjective evolutionary algorithm (MOEA) to solve a very complex network design problem variation called the multicommodity capacitated network design problem (MCNDP). This variation represents a hybrid real-world communication architecture as reflected in real-world network centric models with directional communications, multiple objectives including costs, delays, robustness, vulnerability, and operating reliability within network constraints. Nodes in such centric systems can have multiple and varying link capacities as well as rates and information (commodity) quantities to be sent and received. Each commodity can also have independent prioritized bandwidth and spectrum requirements. The nondominated sorting genetic algorithm (NSGA-II) is selected as the MOEA framework which is modified and parallelized to solve the generic MCNDP. Since the MCNDP is highly constrained but with an enormous number of possible network communication topologies, a novel initialization procedure and mutation method are integrated resulting in reduced search space. Empirical results and analysis indicate that effective topological Pareto solutions are generated for use in highly constrained, communication-based network design.

A Hybrid Simulated Annealing and Simplex Method for Fixed-Cost Capacitated Multicommodity Network Design

The fixed-cost Capacitated Multicommodity Network Design (CMND) problem is a well known NP-hard problem. This paper presents a matheuristic algorithm combining Simulated Annealing (SA) metaheuristic and Simplex method for CMND problem. In the proposed algorithm, a binary array is considered as solution representation and the SA algorithm manages open and closed arcs. Several strategies for opening and closing arcs are proposed and evaluated. In this matheuristic approach, for a given design vector, CMND becomes a Capacitated Multicommodity minimum Cost Flow (CMCF) problem. The exact evaluation of the CMCF problem is performed using the Simplex method. The parameter tuning for the proposed algorithm is done by means of design of experiments approach. The performance of the proposed algorithm is evaluated by solving different benchmark instances. The results of the proposed algorithm show that it is able to obtain better solutions in comparison with previous methods in the literature.

A capacity scaling heuristic for the multicommodity capacitated network design problem

In this paper, we propose a capacity scaling heuristic using a column generation and row generation technique to address the multicommodity capacitated network design problem. The capacity scaling heuristic is an approximate iterative solution method for capacitated network problems based on changing arc capacities, which depend on flow volumes on the arcs. By combining a column and row generation technique and a strong formulation including forcing constraints, this heuristic derives high quality results, and computational effort can be reduced considerably. The capacity scaling heuristic offers one of the best current results among approximate solution algorithms designed to address the multicommodity capacitated network design problem.

On the use of guided design search for discovering significant decision variables in the fixed‐charge capacitated multicommodity network design problem

AbstractA preprocessing technique, Guided Design Search (GDS), is presented for the fixed‐charge multicommodity capacitated network design (FCMD) problem. GDS applies design of experiment (DOE) principles in order to identify the critical edges in FCMD by analyzing the observed effects of tests using strictly generated sets of edge inclusion/exclusion decisions. Reasoning that these critical edges will exist in most of the high‐quality solutions, they are set to their appropriate value, resulting in a reduced solution space for FCMD. High‐quality solutions to this reduced FCMD were found quickly and often improved on those found for the unreduced problem. GDS is shown to dominate several other approaches for reducing the solution space. This success serves to broaden the applicability of experimental design screening and preprocessing techniques. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

0–1 reformulations of the multicommodity capacitated network design problem

We study 0–1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0–1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0–1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and the other based on the addition of extended linking inequalities to the 0–1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities.

A first multilevel cooperative algorithm for capacitated multicommodity network design

We describe the first multilevel cooperative tabu search for the capacitated multicommodity network design problem. Main design challenges are associated to the specification of the problem instance addressed at each level in cooperation, as well as to the definition of the cooperation operators. The paper proposes a first approach to address these challenges and tests it on a set of well-known benchmark problems. The proposed method appears competitive, particularly when difficult problems with many commodities are considered. Directions and challenges for future research are identified and discussed.

Scatter Search for Network Design Problem

A fixed charge capacitated multicommodity network design problem on undirected networks is addressed. At the present time, there exists no algorithm that can solve large instances, common in several applications, in a reasonable period of time. This paper presents an efficient procedure using a scatter search framework. Computational experiments on a large set of randomly generated problems show that this procedure is capable of finding good solutions to large-scale problems within a reasonable amount of time.

Grasp Embedded Scatter Search for the Multicommodity Capacitated Network Design Problem

A GRASP embedded Scatter Search is developed for the multicommodity capacitated network design problem. Difficulty for this problem arises from the fact that selection of the optimal network design...