Capacitated Network Design Research Articles

A fast hybrid primal heuristic for multiband robust capacitated network design with multiple time periods

We investigate the Robust Multiperiod Network Design Problem, a generalization of the Capacitated Network Design Problem (CNDP) that, besides establishing flow routing and network capacity installation as in a canonical CNDP, also considers a planning horizon made up of multiple time periods and protection against fluctuations in traffic volumes. As a remedy against traffic volume uncertainty, we propose a Robust Optimization model based on Multiband Robustness (Büsing and D’Andreagiovanni, 2012), a refinement of classical Γ-Robustness by Bertsimas and Sim that uses a system of multiple deviation bands.Since the resulting optimization problem may prove very challenging even for instances of moderate size solved by a state-of-the-art optimization solver, we propose a hybrid primal heuristic that combines a randomized fixing strategy inspired by ant colony optimization and an exact large neighbourhood search. Computational experiments on a set of realistic instances from the SNDlib show that our original heuristic can run fast and produce solutions of extremely high quality associated with low optimality gaps.

A Lagrangian relaxation approach for expansion of a highway network

This paper deals with the problem of improving an existing road network in the context of strategic planning through the creation of new highway corridors. To address this problem we analyse three mixed integer programming models. The so-called [P1] is the classical capacitated multicommodity network design model. The model named [P2] imposes on [P1] the location of a single (main path) highway corridor in the road network and [P3] adds to [P2] a set of sub-tour breaking constraints. The stated goal is to minimize the total travel time for a known origin-destination demand matrix with a given budget. In this paper we propose an efficient method for [P3], based on a Lagrangian relaxation, to obtain easily-solved sub-problems. A cutting-plane method for solving the Lagrangian sub-problems is proposed. This method generates valid cuts until an optimal solution is found. The Lagrangian dual problem is solved using the sub-gradient optimization method. A case study has been carried out for the region of Castilla-La Mancha (Spain). Computational comparisons between the proposed method and a state-of-the-art mixed-integer code are presented. The Lagrangian relaxation approach is found to be capable of generating good feasible solutions to the case study within a reasonable computational time.

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.

Capacitated Network Design Games

We study a capacitated symmetric network design game, where each of n agents wishes to construct a path from a network's source to its sink, and the cost of each edge is shared equally among the agents using it. The uncapacitated version of this problem has been introduced by Anshelevich et al. (2003) and has been extensively studied. We find that the consideration of edge capacities entails a significant effect on the quality of the obtained Nash equilibria (NE), under both the utilitarian and the egalitarian objective functions, as well as on the convergence rate to an equilibrium. The following results are established. First, we provide bounds for the price of anarchy (PoA) and the price of stability (PoS) measures with respect to the utilitarian (i.e., sum of costs) and egalitarian (i.e., maximum cost) objective functions. Our main result here is that unlike the uncapacitated version, the network topology is a crucial factor in the quality of NE. Specifically, a network topology has a bounded PoA if and only if it is series-parallel (SP), i.e., a network that is built inductively by series compositions and parallel compositions of SP networks. Second, we show that the convergence rate of best-response dynamics (BRD) may take Ω(n1.5) steps. This is in contrast to the uncapacitated version, where convergence is guaranteed within at most n iterations.

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.

Approximability of Capacitated Network Design

In the capacitated survivable network design problem (Cap-SNDP), we are given an undirected multi-graph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimum-cut requirements. Unlike its classical special case of SNDP when all capacities are unit, the approximability of Cap-SNDP is not well understood; even in very restricted settings no known algorithm achieves a o(m) approximation, where m is the number of edges in the graph. In this paper, we obtain several new results and insights into the approximability of Cap-SNDP.We give an O(logn) approximation for a special case of Cap-SNDP where the global minimum cut is required to be at least R. (Note that this problem generalizes the min-cost λ-edge-connected subgraph problem, which is the special case of our problem when all capacities are unit.) Our result is based on a rounding of a natural cut-based LP relaxation strengthened with knapsack-cover (KC) inequalities. Our technique extends to give a similar approximation for a new network design problem that captures global minimum cut as a special case. We then show that as we move away from global connectivity, even for the single pair case (that is, when only one pair (s,t) has positive connectivity requirement), this strengthened LP has Ω(n) integrality gap.We also consider a variant of Cap-SNDP in which multiple copies of an edge can be bought: we give an O(logk) approximation for this case, where k is the number of vertex pairs with non-zero connectivity requirement. This improves upon the previously known O(min{k,logR max})-approximation when R max is large; here R max is the largest requirement. On the other hand, we observe that the multiple copy version of Cap-SNDP is Ω(loglogn)-hard to approximate even for the single-source version of the problem.

A MIP-based approach to solve the prize-collecting local access network design problem

This paper presents a new combinatorial optimization problem that can be used to model the deployment of broadband telecommunications systems in which optical fiber cables are installed between a central office and a number of end-customers. In this capacitated network design problem the installation of optical fiber cables with sufficient capacity is required to carry the traffic from the central office to the end-customers at minimum cost. In the situation motivating this research the network does not necessarily need to connect all customers (or at least not with the best available technology). Instead, some nodes are potential customers. The aim is to select the customers to be connected to the central server and to choose the cable capacities to establish these connections. The telecom company takes the strategic decision of fixing a percentage of customers that should be served, and aims for minimizing the total cost of the network providing this minimum service. For that reason the underlying problem is called the Prize-Collecting Local Access Network Design problem (PC-LAN).We propose a branch-and-cut approach for solving small instances. For large instances of practical importance, our approach turns into a mixed integer programming (MIP) based heuristic procedure which combines the cutting-plane algorithm with a multi-start heuristic algorithm. The multi-start heuristic algorithm starts with fractional values of the LP-solutions and creates feasible solutions that are later improved using a local improvement strategy.Computational experiments are conducted on small instances from the literature. In addition, we introduce a new benchmark set of real-world instances with up to 86,000 nodes, 116,000 edges and 1500 potential customers. Using our MIP-based approach we are able to solve most of the small instances to proven optimality. For more difficult instances, we are not only able to provide high-quality feasible solutions, but also to provide certificate on their quality by calculating lower bounds to the optimal solution values.

A hybrid simulated annealing and column generation approach for capacitated multicommodity network design

This paper presents a hybrid simulated annealing (SA) and column generation (CG) algorithm for the path-based formulation of the capacitated multicommodity network design (PCMND) problem. In the proposed method, the SA metaheuristic algorithm manages open and closed arcs. Several strategies for adding and dropping arcs are suggested and evaluated. For a given design vector in the proposed hybrid approach, the PCMND problem becomes a capacitated multicommodity minimum cost flow (CMCF) problem. The exact evaluation of the CMCF problem is performed using the CG algorithm. The parameter tuning is done by means of design of experiments approach. The performance of the proposed algorithm is evaluated by solving several benchmark instances. The results of the proposed algorithm are compared with the solutions of CPLEX solver and the best-known method in the literature under different time limits. Statistical analysis proves that the proposed algorithm is able to obtain better solutions.

A three-phase matheuristic for capacitated multi-commodity fixed-cost network design with design-balance constraints

This paper proposes a three-phase matheuristic solution strategy for the capacitated multi-commodity fixed-cost network design problem with design-balance constraints. The proposed matheuristic combines exact and neighbourhood-based methods. Tabu search and restricted path relinking meta-heuristics cooperate to generate as many feasible solutions as possible. The two meta-heuristics incorporate new neighbourhoods, and computationally efficient exploration procedures. The feasible solutions generated by the two procedures are then used to identify an appropriate part of the solution space where an exact solver intensifies the search. Computational experiments on benchmark instances show that the proposed algorithm finds good solutions to large-scale problems in a reasonable amount of time.

Capacitated Network Design using Bin-Packing

In this paper, we consider the Capacitated Network Design (CND) problem. We investigate the relationship between CND and the Bin-Packing problem. This is exploited for identifying new classes of valid inequalities for the CND problem and developing a branch-and-cut algorithm to solve it efficiently.

A cutting plane algorithm for the Capacitated Connected Facility Location Problem

We consider a network design problem that arises in the cost-optimal design of last mile telecommunication networks. It extends the Connected Facility Location problem by introducing capacities on the facilities and links of the networks. It combines aspects of the capacitated network design problem and the single-source capacitated facility location problem. We refer to it as the Capacitated Connected Facility Location Problem. We develop a basic integer programming model based on single-commodity flows. Based on valid inequalities for the capacitated network design problem and the single-source capacitated facility location problem we derive several (new) classes of valid inequalities for the Capacitated Connected Facility Location Problem including cut set inequalities, cover inequalities and combinations thereof. We use them in a branch-and-cut framework and show their applicability and efficacy on a set of real-world instances.

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.

A Hierarchical Approach to the Capacitated Single Allocation <i>p</i>-Hub Median Problem with Direct Shipment

In this paper we consider a capacitated single allocation p-hub median problem with direct shipment (CSApHMPwD). We determine the location of p hubs, the allocation of non-hub nodes to hubs, and direct shipment paths in the network. This problem is formulated as 0-1 integer programming model with the objective of the minimum total transportation cost and the fixed cost associated with the establishment of hubs. An optimal solution is found using CPLEX for the small sized problems. Since the CSApHMPwD is NP-hard, it is difficult to obtain optimal solution within a reasonable computational time. Therefore, an ant colony optimization algorithm is developed which solves hub selection and node allocation problem hierarchically. Its performance is examined through a comparative study. The experimental results show that the proposed ant colony optimization algorithm can be a viable solution method for the capacitated hub and spoke 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.

Strategies for protecting supply chain networks against facility and transportation disruptions: an improved Benders decomposition approach

Disruptions rarely occur in supply chains, but their negative financial and technical impacts make the recovery process very slow. In this paper, we propose a capacitated supply chain network design (SCND) model under random disruptions both in facility and transportation, which seeks to determine the optimal location and types of distribution centers (DC) and also the best plan to assign customers to each opened DC. Unlike other studies in the extent literature, we use new concepts of reliability to model the strategic behavior of DCs and customers at the network: (1) Failure of DCs might be partial, i.e. a disrupted DC might still be able to serve with a portion of its initial capacity (2) The lost capacity of a disrupted DC shall be provided from a non-disrupted one and (3) The lost capacity fraction of a disrupted DC depends on its initial investment amount in the design phase.

A Simplex-based simulated annealing algorithm for node-arc capacitated multicommodity network design

The network design is a well-known problem, both of practical and theoretical significance. Network design models are extensively used to represent a wide range of planning and operations management issues in transportation, telecommunications, logistics, production and distribution. This paper presents a solution method for node-arc formulation of capacitated fixed-charge multicommodity network design problems. The proposed method is a hybrid algorithm of Simplex method and simulated annealing metaheuristic. The basic idea of the proposed algorithm is to use a simulated annealing algorithm to explore the solution space, where the revised Simplex method is used to evaluate, select and implement the moves. In the proposed algorithm, the neighborhood structure is pivoting rules of the Simplex method that provide an efficient way to reach the neighbors of current solution. To evaluate the proposed algorithm, the standard problems with different sizes are used. The algorithm parameters are tuned by design of experiments approach and the most appropriate values for the parameters are adjusted. The performance of the proposed algorithm is evaluated by statistical analysis. The results show high efficiency and effectiveness of the proposed algorithm.

Solving train formation problem using simulated annealing algorithm in a simplex framework

SUMMARYThe train formation plan (TFP) determines the train services and their frequencies and assigns the demands. The TFP models are often formulated as a capacitated service network design problem, and the optimal solution is normally difficult to find. In this paper, a hybrid algorithm of the Simplex method and simulated annealing is proposed for the TFP problem. The basic idea of the proposed algorithm is to use a simulated annealing algorithm to explore the solution space, where the revised Simplex method evaluates, selects, and implements the moves. In the proposed algorithm, the neighborhood structure is based on the pivoting rules of the Simplex method that provides an efficient method to reach the neighbors of the current solution. A state‐of‐the‐art method is applied for parameters tuning by using the design of experiments approach. To evaluate the proposed model and the solution method, 25 test problems have been simulated and solved. The results show the efficiency and the effectiveness of the proposed approach. The proposed approach is implemented to develop the TFP in the Iranian railway as a case study. It is possible to save significant time and cost through solving the TFP problem by using the proposed algorithm and developing the efficient TFP plan in the railway networks. Copyright © 2012 John Wiley & Sons, Ltd.

Fractional routing using pairs of failure-disjoint paths

Given a set of commodities and a network where some arcs can fail while others are reliable, we consider a routing problem with respect to a survivability requirement that each commodity can be split among pairs of failure-disjoint paths. Two paths p and p′ form a pair of failure-disjoint paths if they share only reliable arcs. The same flow is sent over p and p′, but the flow sent on a common reliable arc is not doubled.We present a compact linear formulation of the problem. Also three non-compact formulations solvable by column generation are introduced. In the first formulation, the generated columns correspond to pairs of failure-disjoint paths, while in the second formulation the generated columns correspond to simple paths. The third formulation is solved by generating pairs of arc-disjoint paths. All formulations are compared numerically. On top of that we study some generalizations and some special cases of the problem of computing a shortest pair of failure-disjoint paths. One of these generalizations is equivalent to a single-commodity capacitated network design problem.

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.

Progressive hedging‐based metaheuristics for stochastic network design

AbstractWe consider the stochastic fixed‐charge capacitated multicommodity network design (S‐CMND) problem with uncertain demand. We propose a two‐stage stochastic programming formulation, where design decisions make up the first stage, while recourse decisions are made in the second stage to distribute the commodities according to observed demands. The overall objective is to optimize the cost of the first‐stage design decisions plus the total expected distribution cost incurred in the second stage. To solve this formulation, we propose a metaheuristic framework inspired by the progressive hedging algorithm of Rockafellar and Wets. Following this strategy, scenario decomposition is used to separate the stochastic problem following the possible outcomes, scenarios, of the random event. Each scenario subproblem then becomes a deterministic CMND problem to be solved, which may be addressed by efficient specialized methods. We also propose and compare different strategies to gradually guide scenario subproblems to agree on the status of design arcs and aim for a good global design. These strategies are embedded into a parallel solution method, which is numerically shown to be computationally efficient and to yield high‐quality solutions under various problem characteristics and demand correlations. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011.