Hierarchical Network Design Problem Research Articles

Sustainable hierarchical multi-modal hub network design problem: bi-objective formulations and solution algorithms

This paper presents a bi-objective model for the design and optimization of a sustainable hierarchical multi-modal hub network. The proposed model focuses on sustainability by considering economic, environmental, and social aspects of the decisions in a hierarchical network. A case of Turkish network for freight transportation is used to validate the proposed model. To solve the small-sized problems, the augmented epsilon constraint method version 2 (AUGMECON2) is applied. It can be inferred from the Pareto-optimal set obtained by AUGMECON2 that the effect of increasing the number of hubs after a threshold is marginal. The current contribution proposes two multi-objective genetic algorithms (NSGA-II and NRGA), which incorporate LP solving and Dijkstra algorithm. The results show the superiority of NRGA compared to NSGA-II in terms of solution time. Also, we present an alternative, more efficient formulation to the problem. Based on the alternative formulation, in addition to AUGMECON2, we use two exact methods, including Torabi and Hassini (TH) method and augmented weighted Tchebycheff procedure (AWTP), to find Pareto-optimal solutions for small, medium, and large-sized problems (including the case study). The performance of the proposed solution methods is measured using some multi-objective indicators. The results show the superiority of AUGMECON2.

Ring hierarchical hub network design problem: Exact and heuristic solution methods

In this paper, we study the hierarchical hub network design problem with a ring-star-star structure. In this problem, the hubs are located in two layers. In the first layer, central hubs (main hubs) are located in a ring structure, and in the second layer, secondary hubs are located. Each secondary hub is allocated to a central hub. Other demand points can also be allocated to each of these hubs (central hubs or secondary hubs). The objective is to minimize transportation costs on the network. This problem applies to communication networks when establishing direct links between demand nodes is not cost-effective, and there are two levels of service for customers. Also, the ring structure is used to reduce costs associated with full communication between central hubs. We present a mixed-integer programming model for the problem and report the results of the problem solving for a numerical example on the CAB dataset. Also, we present three solution methods for the problem. First, by exploiting the decomposable structure of the proposed model, we introduce an accelerated Benders decomposition algorithm. Then, we present a hybrid genetic algorithm that uses the Dijkstra algorithm to evaluate solutions. Next, we present a hybrid variable neighborhood search algorithm that uses the Dijkstra algorithm to calculate the cost of each solution. Also, a relax-and-fix algorithm is proposed to find near-optimal feasible solutions for the problem. Computational results are presented on the USA423 dataset. The results show that the proposed relax-and-fix algorithm can solve instances with up to 100 nodes. Also, the proposed hybrid algorithms can solve instances with up to 423 nodes. The performance of the proposed hybrid variable neighborhood search is better than the proposed hybrid genetic algorithm in solving large-sized instances. • The paper formulates the hierarchical hub network using a ring structure for the central hubs. • Minimizing the total flow cost while satisfying the demands is considered. • The paper develops solution methodologies based on Accelerated Benders Decomposition, Hybrid Genetic Algorithm, and Hybrid Variable Neighborhood Search Algorithm. • The paper presents computational studies using CAB and USA423 datasets.

A branch-and-price algorithm for the ring/ring problem

The ring/ring problem is a hierarchical network design problem that uses a two-level network, in which each level is a ring with limited capacity. It is related to other hierarchical design problems and also to the vehicle routing problem. Rings are used in telecommunications because of their survivability, while hierarchical networks are useful for large networks. We propose an extended integer programming formulation for the problem and a branch-and-price algorithm to solve it. A relaxation of the pricing problem is considered and also primal and pricing heuristics to improve the performance of the algorithm. Computational experiments were conducted on instances derived from the literature and show the hardness to solve instances with high capacity.

A Multi-Commodity Flow Based Model for Multi Layer Hierarchical Ring Network Design

We address the Multi Layer Hierarchical Ring Network Design Problem. The aim of this problem is to connect nodes that are assigned to different layers using a hierarchy of rings of bounded length. We present a multi-commodity flow based mixed integer linear programming formulation and experimentally evaluate it on various graphs. Instances up to 76 nodes and 281 edges could be solved to optimality.

An evacuation route choice model based on multi-agent simulation in order to prepare Tsunami disasters

ABSTRACTMinimising the damage caused by natural disasters such as tsunamis in vulnerable coastal areas requires the development of an emergency action plan. An evacuation plan can reduce the damaging impact of an unexpected tsunami. One of the most important factors in developing an action plan is a guideline for choosing proper evacuation routes in order to minimise loss of life and property. In this study, an evacuation network is designed to develop an evacuation plan based on a hierarchical network design problem used in communication networks, public transport, and social organisation network designs. Experimental analyses have been carried out in order to find effective evacuation plans at Haeundae Beach, Busan, Korea, using multi-agent transportation simulation (MATSim). In order to design an effective hierarchical evacuation network, a hierarchical evacuation network design is developed using the concepts of hub location, clustering, and network design. Further, we consider contraflow in the evacuation network, as well as determination of corridor lines by the minimum distance rule, maximal flow rule, maximum accessible rule, and minimum travel time rule. As a result of running each scenario using MATSim, a criterion for establishing the corridor line is evaluated. An optimal evacuation route is found to minimise overcrowded queues of evacuees, when the total evacuation time from the area of the tsunami is approximately 3 hours and 30 minutes.

Coordinated dual-homing in designing hierarchical wireless access network with a genetic algorithm based approach

With the growth of mobile users and the increasing deployment of wireless access network infrastructures, the issue of fault tolerance is becoming an important component of efficient wireless access network design. In this work, we study a survivable hierarchical network design problem. Given the available capacity, connectivity, and reliability at each level, the problem is to minimize overall connection cost for multiple requests such that the capacity, connectivity, and minimum survivability constraints are not violated. Our study is different than earlier research in regard to the coordination of multiple layers of access networks. The connectivity to the core network may be fully or partially dual-homed paths, or may be single-homed paths. Dual-homing schemes spanning to different levels in the network hierarchy are used if the single-homed connectivity is not enough to guarantee the minimum required survivability. We formulate the problem using mixed integer linear programming and prove the complexity class to be NP-hard. We then propose an off-line genetic algorithm based meta-heuristic. Given the complexity of the problem, simulation results demonstrate that the proposed approach is viable in designing fault-tolerant access networks with dual-homing capability.

Survivability in hierarchical telecommunications networks

AbstractThe survivable hierarchical telecommunications network design problem consists of locating concentrators, assigning user nodes to concentrators, and linking concentrators in a reliable backbone network. In this article, we study this problem when the backbone is 2‐edge connected and when user nodes are linked to concentrators by a point‐to‐point access network. We formulate this problem as an integer linear program and present a facial study of the associated polytope. We describe valid inequalities and give sufficient conditions for these inequalities to be facet defining. We investigate the computational complexity of the corresponding separation problems. We propose some reduction operations to speed up the separation procedures. Finally, we devise a branch‐and‐cut algorithm based on these results and present the outcome of a computational study. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

케이블 비용을 고려한 링메쉬 구조의 광통신망 설계

In this paper, we deal with a hierarchical ring-mesh optical network design problem. The objective is to minimize the total cost of optical add-drop multiplexers (OADMs) handling intra-ring traffic, optical cross-connects (OXCs) handling inter-ring traffic, and cabling cost among OADMs and among OXCs, while satisfying intra-ring and inter-ring capacities. We develop an integer programming (IP) formulation for the problem and devise some cutting planes that partially break the symmetry of rings. Dealing with the inherent computational complexity of the problem, we devise an effective heuristic procedure that finds a good quality feasible solution within reasonable computing times. Computational results demonstrate the efficacy of the proposed solution procedure; the developed symmetry breaking inequalities significantly reduce the computing time to find an optimal solution for small size problems, and the heuristic procedure finds a better feasible solution than that CPLEX, a commercial optimization software, finds for large size problems.

A branch and cut algorithm for the hierarchical network design problem

The Hierarchical Network Design Problem consists of locating a minimum cost bi-level network on a graph. The higher level sub-network is a path visiting two or more nodes. The lower level sub-network is a forest connecting the remaining nodes to the path. We optimally solve the problem using an ad hoc branch and cut procedure. Relaxed versions of a base model are solved using an optimization package and, if binary variables have fractional values or if some of the relaxed constraints are violated in the solution, cutting planes are added. Once no more cuts can be added, branch and bound is used. The method for finding valid cutting planes is presented. Finally, we use different available test instances to compare the procedure with the best known published optimal procedure, with good results. In none of the instances we needed to apply branch and bound, but only the cutting planes.

Hierarchical parcel distribution facility ground network design problem

This paper is concerned with the network design problem for parcel ground distribution facilities. The design involves three interacting levels of facilities in addition to customers, with each lower level being assigned to exactly one higher level. The link cost, depending on the transportation mode, is expressed as a function of the link load, which is determined by the assignment variables. The resulting model is a complex integer program with non-linear objective function. The model is solved by a heuristic method where the lower-level solution provides input data for the higher-level one. The method is illustrated with an example based on a real world application.

Hierarchical spanning tree network design with Nash genetic algorithm

In this paper, we will propose a Nash genetic algorithm (Nash GA) for solving a hierarchical spanning tree network design problem, formulated as a bi-level programming problem. The proposed algorithm can be employed in designing the backbone topology in a hierarchical link-state (LS) routing domain. Because the well-designed backbone topology structure has a great impact on the overall routing performance in a hierarchical LS domain, the importance of this research is evident. The proposed algorithm is to find an optimal configuration of backbone network for backbone provider (BP) and distribution network for internet service provider (ISP), properly meeting two-aspect engineering goals: i.e., average message delay and connection costs. Also, it is assumed that there are the decision makers for BP and the decision makers for ISP join in the decision making process in order to non-cooperatively optimize the own objective function. From the experiment results, we can see clearly that our proposed algorithm can be employed in effectively designing the spanning tree network of hierarchical LS routing domain considering not only engineering aspects but also specific benefits from systematical layout of backbone network.

An optimal procedure for solving the hierarchical network design problem

The hierarchical network design problem consists of finding a minimum cost bilevel network that connects all the nodes in a set, created by a loopless main path and a forest. The main path is formed by primary (higher cost) arcs, providing a path between an origin node and a destination node. The forest, built using secondary (lower cost) arcs, connects all the nodes not on the main path, to the path itself. We state and prove some properties of the problem, which allow finding good upper bounds to the solution in polynomial time when the primary costs are proportional to secondary costs. We also propose an O(n 4) procedure to improve on these bounds. In turn, these bounds are used to significantly reduce the number of nodes and arcs of the problem. Once the problem is reduced, large instances can be solved to optimality. At this stage, we use one of two linear integer optimization formulations. The first and preferred one is based on multicommodity flows, which avoids the formation of subtours. The second formulation avoids subtours by iteratively adding ad hoc constraints. We show some examples and provide computational experiments performed on networks with sizes up to 600 nodes and 14 000 arcs.

The hierarchical network design problem for time-definite express common carriers

Time-definite express common carriers provide time guaranteed door-to-door express service for small parcel shipments. Centers pick up and deliver parcels, while hubs consolidate partial loads. Each center is connected through a secondary route to its primary hub, while hubs are mutually connected by primary routes in a hierarchical hub-and-spoke network. The carriers may dispatch large trucks/aircraft on the primary routes but utilize smaller trucks/aircraft on the secondary routes. The time-constrained hierarchical hub-and-spoke network design problem involves determining the fleet size and schedules on the primary and secondary routes to minimize the total operating cost, while satisfying the desired level of service. We developed a route-space directed network and modeled the problem as a 0–1 binary program. An implicit enumeration method with an embedded least time path subproblem was developed. The sensitivity analysis on the service level in a partial line-haul operations network for the second largest carrier in Taiwan showed that the costs are not strictly monotonically increasing with the service levels, rather they are monotonically non-decreasing according to a step function. In addition, the determination of the sort start and pickup cutoff times has a great impact on the total cost.

The hierarchical network design problem with multiple primary paths

In this paper we develop a suboptimal solution for the hierarchical network design problem (HNDP) with multiple primary paths. In practice some networks may require solutions to the HNDP with more than one origin/destination pair to be connected by primary paths. This paper develops a dynamic programming solution to find a suboptimal solution to the problem.

An addendum to the hierarchical network design problem

The hierarchical network design problem is the problem to find a spanning tree of minimum total weight, when the edges of the path between two given nodes are weighted by an other cost function than the tree edges not on this path. We point out that a dynamic programming oriented heuristic can already be found in literature. Further we report on possible extensions and improvements.

The Vehicle Routing-Allocation Problem: A unifying framework

In this paper the Vehicle Routing-Allocation Problem (VRAP) is presented. In VRAP not all customers need be visited by the vehicles. However customers not visited either have to be allocated to some customer on one of the vehicle tours or left isolated. We concentrate our discussion on the Single Vehicle Routing-Allocation Problem (SVRAP). An integer linear programming formulation of SVRAP is presented and we show how SVRAP provides a unifying framework for understanding a number of the papers and problems presented in the literature. Specifically the covering tour problem, the covering salesman problem, the median tour problem, the maximal covering tour problem, the travelling salesman problem, the generalised travelling salesman problem, the selective travelling salesman problem, the prize collecting travelling salesman problem, the maximum covering/shortest path problem, the maximum population/shortest path problem, the shortest covering path problem, the median shortest path problem, the minimum covering/shortest path problem and the hierarchical network design problem are special cases/variants of SVRAP.

Modeling and Heuristic Worst-Case Performance Analysis of the Two-Level Network Design Problem

This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. We are given an undirected network containing two types of nodes—primary and secondary—and fixed costs for installing either a primary or a secondary facility on each edge. Primary nodes require higher grade interconnections than secondary nodes, using the more expensive primary facilities. The TLND problem seeks a minimum cost connected design that spans all the nodes, and connects primary nodes via edges containing primary facilities; the design can use either primary or secondary edges to connect the secondary nodes. The TLND problem generalizes the well-known Steiner network problem and the hierarchical network design problem. In this paper, we study the relationship between alternative model formulations for this problem (e.g., directed and undirected models), and analyze the worst-case performance for a composite TLND heuristic based upon solving embedded subproblems (e.g., minimum spanning tree and either Steiner tree or shortest path subproblems). When the ratio of primary to secondary costs is the same for all edges and when we solve the embedded subproblems optimally, the worst-case performance ratio of the composite TLND heuristic is 4/3. This result applies to the hierarchical network design problem with constant primary-to-secondary cost ratio since its subproblems are shortest path and minimum spanning tree problems. For more general situations, we express the TLND heuristic worst-case ratio in terms of the performance of any heuristic used to solve the embedded Steiner tree subproblem. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of a multi-level extension of this problem.

A Dual-Based Algorithm for Multi-Level Network Design

Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels or grades, the Multi-level Network Design (MLND) problem seeks a fixed cost minimizing design that spans all the nodes and connects the nodes at each level by facilities of the corresponding or higher grade. This problem generalizes the well-known Steiner network problem and the hierarchical network design problem, and has applications in telecommunication, transportation, and electric power distribution network design. In a companion paper we studied alternative model formulations for a two-level version of this problem, and analyzed the worst-case performance of several heuristics based on Steiner network and spanning tree solutions. This paper develops a dual-based algorithm for the MLND problem. The method first performs problem preprocessing to fix certain design variables, and then applies a dual ascent procedure to generate upper and lower bounds on the optimal value. We report extensive computational results on large, random two-level test problems (containing up to 500 nodes, and 5,000 edges) with varying cost structures. The integer programming formulation of the largest of these problems has 20,000 integer variables and over 5 million constraints. Our tests indicate that the dual-based algorithm is very effective, producing solutions that are within 0.9% of optimality.

The Hierarchical Network Design Problem: A New Formulation and Solution Procedures

In this paper we develop a new heuristic for the hierarchical network design problem. The heuristic is based upon Lagrangian relaxation of a reformulation of the problem. In addition, we develop a branch and bound algorithm to identify the optimal solution for the 5% of the test problems for which the new heuristic failed to do so. This algorithm proved to be very efficient due to the tight bounds generated by the heuristic.

The multi-weighted Steiner tree problem

We formulate and investigate the Multi-Weighted Steiner Problem (MWS), a generalization of the Steiner problem in graphs, involving more than one weight function. As a special case, it contains the hierarchical network design problem. With the notion of "bottleneck length/distance", a min-max measure, we analyze the interaction between differently weighted edges in a solution. Combining the results with known methods for the Steiner problem in graphs and the hierarchical network design problem, two heuristics for the MWS are developed, one based on weight modifications and the other on exchanging edges. Both are of time complexityO(kv2), withv the number of nodes andk the number of special nodes in the graph. The first is also suited for thedirected MWS; the second is expected to perform better on the undirected version. Before actually solving the Steiner problem in graphs and the hierarchical network design problem, preprocessing techniques exploiting tests to reduce the problem graphs have proven to be valuable. We adapt three prominent tests for use in the MWS.