Capacitated Network Design Research Articles

Identifying critical demand scenarios for the robust capacitated network design problem using principal component analysis

AbstractIn this paper, we consider the single‐commodity robust network design problem. Given an undirected graph with capacity installation costs on its edges and a set of scenarios with associated flow balance vectors that represent different scenarios of node supplies and demands, the goal is to find integer edge capacities that minimize the total installation cost and permit a feasible single commodity flow for each scenario. This problem arises, for example, in the design of power networks, which are dimensioned to accommodate many different load scenarios. We propose a new method to identify a small subset of the given scenarios, such that solving the robust network design problem for the smaller scenario set leads to almost the same capacities as solving it for the full scenario set . By considering only the scenarios in , the size of the model that needs to be solved can be reduced substantially, while the error introduced by neglecting the remaining scenarios is kept very small. Our method only employs simple techniques from statistical data analysis, namely principal component analysis (PCA), and convex hull computations in low dimensions. Thus, its computational effort is very small and it is easily applicable to more complex network design problems. We evaluate the effectiveness of the method in computational experiments for instances stemming from offshore power grid planning and telecommunication networks. Our results show that the proposed techniques are indeed well suited to identify small scenario subsets that lead to significantly reduced models with high quality solutions.

Lagrangean relaxation algorithms for fixed-charge capacitated relay network design

We consider a fixed-charge capacitated relay network design problem (CRNDP), which has widespread applications in the design of telecommunication and transportation networks, where relay points are presented as signal regenerators, truckload driver and/or tractor exchange stations or electrical vehicle recharging station. Our CRNDP integrates the key features of single allocation hub location problem (SAHLP) and capacitated network design problem (CNDP). Typical range restrictions in truckload transportation, electrical vehicles as well as for keeping signal strength in telecommunications, and further limitations of flow on the links of an underlying network in these mediums motivate the use of distance constraints and link capacity constraints in the design of a relay network for this diverse set of applications. To efficiently solve our model, we propose a Lagrangean relaxation algorithm, where the upper bounds are enhanced by a heuristic during the iterations and the lower bound is improved by the introduction of surrogate constraints. The extensive computational studies are conducted to examine the performance of the algorithm and the results show that the solution method we propose can efficiently solve CRNDP and obtain good-quality solutions, even for very large size instances.

Framework for optimization of long-term, multi-period investment planning of integrated urban energy systems

In order to achieve stringent greenhouse gas emission reductions, a transition of our entire energy system from fossil to renewable resources needs to be designed. Such an energy transition brings two main challenges: most renewables generate variable electric energy, yet most demand is currently not electric (carrier mismatch) and does not always manifest at the same time as supply (temporal mismatch). Integrating multiple energy infrastructures can address both challenges by using the synergy between different energy carriers; building on existing infrastructure, while allowing a robust and flexible integration of the new.This paper proposes an optimization framework for long-term, multi-period investment planning of urban energy systems in an integrated manner. We formulate it as a mixed-integer linear program, combining a capacitated facility location with a multi-dimensional, capacitated network design problem. It includes generation and network expansion planning as well as interconnections between networks and storage infrastructure for each energy system. It can incorporate pathway effects like techno-economic developments, policy measures, and weather variations. The intended use is to support urban decision makers with long-term investment planning, though it can be tailored to fit other geographical or temporal scales.We demonstrate the model using two cases based on an average city in The Netherlands, which wants to reduce its CO2-emissions with 95% by 2050. In the first case, we include explicit carbon-emission constraints to study the effects of the carrier mismatch. In the second case, we implement interannual weather variations to analyze the temporal mismatch. The results give valuable insights into the energy transition design strategy for urban decision makers. They also show the future potential, as well as the computational challenges of the optimization framework.

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.

An exact separation algorithm for unsplittable flow capacitated network design arc-set polyhedron

In this paper, we concentrate on generating cutting planes for the unsplittable capacitated network design problem. We use the unsplittable flow arc-set polyhedron of the considered problem as a substructure and generate cutting planes by solving the separation problem over it. To relieve the computational burden, we show that, in some special cases, a closed form of the separation problem can be derived. For the general case, a brute-force algorithm, called exact separation algorithm, is employed in solving the separation problem of the considered polyhedron such that the constructed inequality guarantees to be facet-defining. Furthermore, a new technique is presented to accelerate the exact separation algorithm, which significantly decreases the number of iterations in the algorithm. Finally, a comprehensive computational study on the unsplittable capacitated network design problem is presented to demonstrate the effectiveness of the proposed algorithm.

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.

Lagrangian bounds for large‐scale multicommodity network design: a comparison between Volume and Bundle methods

AbstractThe Bundle Method and the Volume Algorithm are among the most efficient techniques to obtain accurate Lagrangian dual bounds for hard combinatorial optimization problems. We propose here to compare their performance on very large scale Fixed‐Charge Multicommodity Capacitated Network Design problems. The motivation is not only the quality of the approximation of these bounds as a function of the computational time but also the ability to produce feasible primal solutions and thus to reduce the gap for very large instances for which optimal solutions are out of reach. Feasible solutions are obtained through the use of Lagrangian information in constructive and improving heuristic schemes. We show in particular that, if the Bundle implementation has provided great quality bounds in fewer iterations, the Volume Algorithm is able to reduce the gaps of the largest instances, taking profit of the low computational cost per iteration compared to the Bundle Method.

Hallucination Helps: Energy Efficient Virtual Circuit Routing

We consider virtual circuit routing protocols with an objective of minimizing energy in a network of components that are speed scalable, and that may be shut down when idle. We assume the standard model for component power: the power consumed by a component with load (speed) $s$ is $\sigma+ s^\alpha$, where $\sigma$ is the static power and the exponent $\alpha>1$. We obtain a very simple $O(\log^\alpha k)$-approximation algorithm for multicommodity routing, where $k$ is the number of demand pairs. This improves upon previous results by several logarithmic factors. The key step in our algorithm is a random sampling technique that we call hallucination, which is reminiscent of the sample-augment framework for buy-at-bulk problems, and sampling in cut-sparsification algorithms. We also consider the online setting of the problem, where demand pairs arrive over time. We show that our offline algorithm naturally extends to the online setting, and obtain a randomized competitive ratio of $\tilde{O}( \log^{3\alpha + 1} k)$, which is the first nontrivial bound. The analysis of this algorithm involves the study of priority multicommodity flows, where edges and demand-pairs have priorities and each demand-pair must route its flow only on edges of lower priority. We establish a polylogarithmic flow-cut gap for these priority flows, which we believe is of independent interest. Finally, we show how our technique can be used to achieve a randomized $( O(\log m), O(\log^2 m))$ bicriteria competitive algorithm for the uniform capacitated network design problem, where $m$ is the number of edges. Here, every edge has a cost $c_e$ and uniform capacity $q$, and the goal is to choose the minimum cost subgraph that can support the given multicommodity demand. This is the first online algorithm for this problem. In fact, our approach also improves prior results in the offline setting by several logarithmic factors.

Cell-Based Local Search Heuristics for Guide Path Design of Automated Guided Vehicle Systems With Dynamic Multicommodity Flow

This article discusses the guide path design of automated guided vehicle (AGV) systems for which we propose a model that incorporates a dynamic multicommodity flow with capacity constraints from a pickup to a delivery point. The problem is formulated as the selection of a guide path connecting a given set of pickup/delivery points. A cell-based local search that is based on an original neighborhood technique is developed. In the proposed method, the guide path is effectively optimized based on a local search of cell-based neighborhood search preserving connectivity constraints and the subsequent solution of a dynamic multicommodity flow problem. The performance of the proposed method is strengthened by the redundant elimination of arcs and intensification of search according to the concept of the flow-concentrated cell. The effectiveness of the proposed method is demonstrated by comparing it with a general-purpose solver and recent algorithms for fixed-charge capacitated multicommodity network design problems. A real case study is presented to demonstrate the applicability of the proposed method by using simulation software. Note to Practitioners —Due to the recent growth of online shipping, the design of automated warehousing systems is receiving much attention. Most automated warehousing systems introduce multiple automated guided vehicles (AGVs) for transportation. This article presents an efficient optimization of guide path design considering congestions and dynamic multicommodity flow routing. We develop cell-based local search heuristics for the guide path design problem for AGV systems that can be applied to a large-scale transportation model to solve the problem efficiently. Our major finding is that the guide path design with the dynamic multicommodity flow can significantly reduce the delays caused by congestions and improve the efficiency of the warehousing systems. The effectiveness of the derived guide path design is evaluated by simulation software. The feasibility of the guide path is confirmed in several case studies.

Capacitated Multi-Layer Network Design with Unsplittable Demands: Polyhedra and Branch-and-Cut

We consider the Capacitated Multi-Layer Network Design with Unsplittable demands (CMLND-U) problem. Given a two-layer network and a set of traffic demands, this problem consists in installing minimum cost capacities on the upper layer so that each demand is routed along a unique “virtual” path (even using a unique capacity on each link) in this layer, and each installed capacity is in turn associated a “physical” path in the lower layer. This particular hierarchical and unsplittable requirement for routing arises in the design of optical networks, including optical OFDM based networks. In this paper, we give an ILP formulation to the CMLND-U problem and we take advantage of its sub-problems to provide a partial characterization of the CMLND-U polytope including several families of facets. Based on this polyhedral study, we develop a Branch-and-Cut algorithm for the problem and show its effectiveness though a set of experiments, conducted on SNDlib-derived instances and also on real instances.

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.

Increased flexibility in multi echelon multi capacitated supply chain network design

The multi echelon, multi capacitated supply chain network design challenge is to determine the numbers, locations and capacity levels of plants and warehouses; as well as the product flow from plants to warehouses and then from warehouses to customer zones in order to meet demand at minimum cost. Mathematical models for multi capacitated supply chain network design provide a finite set of capacity levels from which to choose; and include variables and constraints to ensure the selection of a single capacity level for each facility to be built. By eliminating the constraints that enforce a single capacity selection, we allow for the selection of several capacity levels for a single plant or warehouse. If such a selection occurs, the plant or warehouse is built with size equal to the sum of the selected capacity levels. This gives an exponential increase in the number of available capacity levels. The increased flexibility allows for less costly supply chain network designs. We present numerical results that demonstrate improved solutions, that is, lower cost supply chain network designs, with lower computational effort.

Increased flexibility in multi echelon multi capacitated supply chain network design

The multi echelon, multi capacitated supply chain network design challenge is to determine the numbers, locations and capacity levels of plants and warehouses; as well as the product flow from plants to warehouses and then from warehouses to customer zones in order to meet demand at minimum cost. Mathematical models for multi capacitated supply chain network design provide a finite set of capacity levels from which to choose; and include variables and constraints to ensure the selection of a single capacity level for each facility to be built. By eliminating the constraints that enforce a single capacity selection, we allow for the selection of several capacity levels for a single plant or warehouse. If such a selection occurs, the plant or warehouse is built with size equal to the sum of the selected capacity levels. This gives an exponential increase in the number of available capacity levels. The increased flexibility allows for less costly supply chain network designs. We present numerical results that demonstrate improved solutions, that is, lower cost supply chain network designs, with lower computational effort.

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.

A Branch-and-Cut algorithm for the Capacitated Multi-Failure Survivable Network Design problem

Abstract Telecommunication networks can be seen as the stacking of several layers like, for instance, IP-over-Optical networks. This infrastructure should have sufficient capacities to route some demands between their origin-destination nodes. In this paper we consider the Capacitated Multi-Failure Survivable Network Design problem. We study two variants of this problem with simple and multiple capacities. We give two multicommodity flow formulations for each variant of this problem and describe some valid inequalities. In particular, we characterize valid inequalities obtained using Chvatal-Gomory procedure from the well known Cutset inequalities. We show that some of these inequalities are facet defining. We discuss separation routines for all the valid inequalities. Using these results, we develop a Branch-and-Cut algorithm and a Branch-and-Cut-and-Price algorithm for each variant and present extensive computational results.

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.

Approximation Schemes for Capacitated Geometric Network Design

We study a capacitated network design problem in a geometric setting. The input consists of an integral edge capacity k and two sets of points on the Euclidean plane, sources, and sinks, with an integral demand for each point. The demand of each source specifies the amount of flow that has to be shipped from the source, and the demand of each sink specifies the amount of flow that has to be shipped to the sink. The goal is to construct a minimum-length network that allows one to route the requested flow from the sources to the sinks and where each edge in the network has capacity k. The vertices of the network are not constrained to the sets of sinks and sources-any point on the Euclidean plane can be used as a vertex. The flow is splittable and parallel edges are allowed. The capacitated geometric network design problem generalizes, among others, the geometric Steiner tree problem, and as such it is NP-hard. We show that if the demands are polynomially bounded and the edge capacity k is not too large, the single-sink capacitated geometric network design problem admits a polynomial time approximation scheme. If the capacity is arbitrarily large, then we design a quasi-polynomial time approximation scheme for the capacitated geometric network design problem allowing for an arbitrary number of sinks. Our results rely on a derivation of an upper bound on the number of vertices different from sources and sinks (the so-called Steiner vertices) in an optimal network. The bound is polynomial in the total demand of the sources. (Less)

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.