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Journal ArticleDOI

A branch-and-cut algorithm for two-level survivable network design problems

TL;DR: This paper approaches the problem of designing a two-level network protected against single-edge failures by giving integer programming formulations and valid inequalities for the different versions of the problem, solving them using a branch-and-cut algorithm, and discussing computational results.
About: This article is published in Computers & Operations Research.The article was published on 2016-03-01 and is currently open access. It has received 16 citations till now. The article focuses on the topics: Backbone network & Network planning and design.

Summary (3 min read)

1. Introduction

  • The authors study several two-level network design problems with survivability requirements in both levels.
  • The lower level networks are called access networks and they connect the users to hubs.
  • There are also studies on two-level networks with survivability requirements on the backbone network.
  • They analyze worst-case performances of some heuristics for some special cases.

2. MIP models

  • Note that this is not a restrictive assumption as one can solve the problem for different root nodes if one has not been a priori fixed.
  • If an edge is used in the backbone network then, due to constraints (5) and (6), both endpoints should be hubs.
  • Finally, constraints (10) and (11) are variable restrictions.
  • In particular, to model the 2EC/ring and ring/ring design problems the authors add the degree constraints xðδðiÞÞr2 8 iAV : ð12Þ.
  • These constraints limit to at most two the degree of a node in an access network and, together with the connectivity constraints (8), force those networks to be rings.

3. Valid inequalities

  • This section presents several families of valid inequalities for the four variants of the 2-LSNDP.
  • The feasible set of this problem is a relaxation of the 2EC/2EC network design problem obtained by dropping the constraints related with xe variables.
  • The authors now present three new families of inequalities.
  • There are at least rðSÞ PiA Szii distinct hubs in set V⧹S that serve nodes in S. Hence, overall at least 2 rðSÞ PiA SziiþPmk ¼ 1 zðjk : SkÞ access edges cross the cut δðSÞ.□ ð21Þ to solve the hub location routing problem.

4. Branch-and-cut algorithm

  • The authors propose an exact branch-and-cut algorithm to solve the 2- LSNDP based on the MIP model strengthened with the valid inequalities presented in the previous section.
  • The branch-and-cut approach consists of a cutting plane technique embedded into a branch-and-bound framework.
  • The authors describe next the main features of the algorithm.

4.1. Initialization

  • To start the optimization the authors solve the linear program (LP) given by the constraints (2)–(6) and variable bounds.
  • To these inequalities, the authors add the degree inequalities (12) when the access networks are required to be rings, i.e., when the variant to solve is the 2EC/ring or the ring/ring.
  • Similarly, to solve the design problems with required ring structure in the backbone network (i.e., ring/ 2EC and ring/ring) the authors add constraints (13).
  • When the backbone network does not need to be a ring, the authors replace (13) by (7) with S¼ fig for all iAV in the initial LP.

4.2. Cutting plane phase

  • If the optimal solution of the LP relaxation is integer, the authors check whether it is a feasible solution for the 2-LSNDP by applying the separation routines for constraints (14), (7) and (8).
  • The cutting plane phase is performed only each 10 branch-and-cut nodes.
  • The left hand side of all these inequalities is the same and they do not involve variables related to the backbone network.
  • The authors refer to the variable values of the current fractional solution by ðxn; yn; znÞ.

4.2.1. Separation of inequalities (7)

  • To separate constraints (7) the authors use an exact and polynomial procedure presented in [20], and inspired by the known separation algorithm for the subtour elimination constraints for the travelling salesman problem.
  • The procedure consists of solving max-fow/min-cut problems on an appropriately defined support graph.
  • Next the authors determine a min-cut set S V 0 with iAS and 0=2S, and finally they check the violation of inequality (7) for that set.

4.2.3. Separation of inequalities (14)

  • Inequalities (14) can be separated in polynomial time.
  • If the inequality for this choice of S is not violated, then there exists no violated inequality (14) for edge fi; jg.

4.2.4. Separation of inequalities (15)

  • Constraints (15) are heuristically separated using a procedure presented in [11] that works as follows.
  • The authors look for sets of nodes fv1;…; vpg that determine odd cycles in G0.
  • The set F is formed by taking an odd number of edges from δðV0Þ with fractional values yne41=2.
  • Then, the corresponding inequality is checked for violation.

4.2.5. Separation of inequalities (16)

  • The bridge inequalities (16) can be separated exactly in polynomial time with the following algorithm.
  • If the capacity of this cut is less than 2yne , the corresponding inequality (16) is violated.
  • Therefore the authors use the following strategy to reduce the number of min-cut computations.
  • Note that, with these changes, the separation procedure for inequalities (16) becomes heuristic.

4.2.6. Separation of inequalities (18)

  • The capacity of the edges in E0 is set to the positive value considered for their definition.
  • Therefore, again the separation problem can be solved exactly by performing a min-cut computation for each pair of nodes.

4.2.7. Separation of inequalities (22)

  • To separate constraints (22) the authors use a heuristic procedure based on the separation algorithm for (20) described in [27].
  • The edges in E0 are the edges in E, with capacity xne ; the edges connecting s with each iAV , with capacity 2ðznjiþ1=qÞ; and the edges connecting each iAV with t, with capacity 2znii.

4.2.8. Separation of inequalities (24)

  • This section describes a heuristic separation for (24) based on an exact and polynomial-time separation for a closely related family of inequalities.
  • These inequalities are the ones obtained by not considering the rounding up operator in (24), i.e. yðEðSÞÞþyðδðSÞÞþxðδðSÞÞZ.
  • This separation algorithm can be adapted to deal with other inequalities (23) with m¼1, like for example when r(S) is defined as in (21).

4.3. Branching strategy

  • The branching strategy the authors devised prioritizes the set of variables zjj.
  • This is done because fixing the set of hubs contributes to determine other features of the solution.
  • Branching on the other variables is done only when all variables zjj are integral.

5. Computational results

  • The authors used CPLEX 12.5 as mixed integer linear programming solver.
  • For a given number of nodes n and fix hub cost setting f j, the hardest instances are those with tighter capacity.
  • To assess the influence of the different sets of valid inequalities proposed in this work, the authors compare four versions of the branchand-cut algorithm that differ by the sets of valid inequalities considered.
  • The addition of constraints (18) reduces the gaps and the running times, this being more evident for Class II instances.
  • This happens because the violation of constraints (17) is checked inside the separation routines for (8) and (18), and the authors are certain to find a violated constraint of the first type when any of the two latter is violated.

6. Conclusions

  • This paper addresses a two-level survivable network design problem in which location, assignment, and network design tasks are jointly tackled.
  • The authors present an integer programming formulation and valid inequalities for the problem.
  • This formulation models, with minor modifications, the four possible combinations of the two singleedge protected topologies considered (rings and two-edge connected networks).
  • The authors have designed an exact branch- and-cut algorithm and they have tested its performance on two sets of instances with different degree of difficulty.
  • The results are satisfactory since they show that instances with up to 40 nodes, which is already a large size for this kind of problems, are solved to optimality in a reasonable amount of time.

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Citations
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Journal ArticleDOI
TL;DR: A goal programming-based multi-objective artificial bee colony optimization (MOABC) algorithm to solve the problem of topological design of distributed local area networks (DLANs) and results indicate that EMOABC demonstrated superior performance than all the other algorithms.
Abstract: The topological design of a computer communication network is a well-known NP-hard problem. The problem complexity is further magnified by the presence of multiple design objectives and numerous design constraints. This paper presents a goal programming-based multi-objective artificial bee colony optimization (MOABC) algorithm to solve the problem of topological design of distributed local area networks (DLANs). Five design objectives are considered herein, namely, network reliability, network availability, average link utilization, monetary cost, and network delay. Goal programming (GP) is incorporated to aggregate the multiple design objectives into a single objective function. A modified version of MOABC, named as evolutionary multi-objective ABC (EMOABC) is also proposed which incorporates the characteristics of simulated evolution (SE) algorithm for improved local search. The effect of control parameters of MOABC is investigated. Comparison of EMOABC with MOABC and the standard ABC (SABC) shows better performance of EMOABC. Furthermore, a comparative analysis is also done with non-dominated sorting genetic algorithm II (NSGA-II), Pareto-dominance particle swarm optimization (PDPSO) algorithm and two recent variants of decomposition based multi-objective evolutionary algorithms, namely, MOEA/D-1 and MOEA/D-2. Results indicate that EMOABC demonstrated superior performance than all the other algorithms.

47 citations


Cites background or methods from "A branch-and-cut algorithm for two-..."

  • ...[5] used a branch-and-cut algorithm for two-level network design for protection against single edge failures....

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  • ...The design objectives considered in these studies include cost [6,7,10,19,2,5,4,16,15,12,20,18], network delay [1,17,8], reliability [13,14,3], blocking probability [11], and total registration signalling [9]....

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Journal ArticleDOI
TL;DR: In this paper, the authors proposed a branch-and-cut solution method to solve the problem of finding the optimal routes for emergency vehicles considering the length, the travel time and the number of paths as performance metrics of network vulnerability.
Abstract: Since disasters have considerable effects on transportation networks, the functionality of an emergency transportation network can play an important role in mitigation phase, especially in developing countries that sometimes suffer the sad experience of almost complete destruction of several cities. Transportation related disaster response activities typically include search and rescue, emergency medical care and fire-fighting trips. In this paper, the emergency transportation network design problem is proposed to determine the optimal network to perform emergency response trips with high priority in the aftermath of earthquakes. The problem has three objective functions designated to identify the optimal routes for emergency vehicles considering the length, the travel time and the number of paths as performance metrics of network vulnerability. A combined approach for considering the three objectives including weighted sum and lexicographic methods is used. The proposed model is solved using a branch-and-cut solution method. The suggested method is tested on the well-known Sioux-Falls network as well as on the real-world network of Tehran metropolis, Iran. Computational experiments are conducted to examine the effects of varying the maximum network length, and the relative weights of other objectives.

42 citations

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TL;DR: New mixed integer linear programming models for the p-Median problem subject to ring, tree and star backbone topology constraints on the facility locations are proposed and variable neighborhood search (VNS) meta-heuristic algorithms are proposed, one for each topology.

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TL;DR: This paper formulated the DLAN topology design problem as a multi-objective optimization problem considering five design objectives and formulated the proposed fuzzy goal programming-based ant colony optimization algorithm (GPACO), which was able to find solutions of higher quality.
Abstract: Topology design of a distributed local area network (DLAN) is a complex optimization problem and has been generally modelled as a single-objective optimization problem. Traditionally, iterative techniques such as genetic algorithms and simulated annealing have been used to solve the problem. In this paper, we formulated the DLAN topology design problem as a multi-objective optimization problem considering five design objectives. These objectives are network reliability, network availability, average link utilization, monetary cost, and average network delay. The multi-objective nature of the problem has been addressed by incorporating a fuzzy goal programming approach to combine the individual design objectives into a single-objective function. The objective function is then optimized using the ant colony algorithm adapted for the problem. The performance of the proposed fuzzy goal programming-based ant colony optimization algorithm (GPACO) is evaluated with respect to the algorithm control parameters, namely pheromone deposit and evaporation rate, colony size and heuristic values. A comparative study was also done using four other multi-objective optimization algorithms which are non-dominated sorting genetic algorithm II, archived multi-objective simulated annealing algorithm, lexicographic ant colony optimization, and Pareto-dominance ant colony optimization. Results revealed that, in general, GPACO was able to find solutions of higher quality as compared to the other four algorithms.

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TL;DR: This paper introduces the concept of system survivability under attack in analogy with system reliability, and provides the expected number of attacks for each system configuration based on the particular attack strategy both for single and multiple attacks.

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References
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Journal ArticleDOI
TL;DR: The ring spur assignment problem (RSAP) is described, a new problem arising in the design of next generation networks and a branch-and-cut decomposition heuristic algorithm suitable for solving problem instances in a reasonable time is described.

9 citations


"A branch-and-cut algorithm for two-..." refers background in this paper

  • ...Carroll and McGarraghy [3] also propose to decompose the problems of designing the rings in different levels....

    [...]

Journal ArticleDOI
TL;DR: This study considers the design of a two-level survivable telecommunications network and gives alternative formulations using cut inequalities, compares these formulations, provides a polyhedral analysis of the small-sized formulation, and describes valid inequalities.
Abstract: The motivation behind this study is the essential need for survivability in the telecommunications networks. An optical signal should find its destination even if the network experiences an occasional fiber cut. We consider the design of a two-level survivable telecommunications network. Terminals compiling the access layer communicate through hubs forming the backbone layer. To hedge against single link failures in the network, we require the backbone subgraph to be two-edge connected and the terminal nodes to connect to the backbone layer in a dual-homed fashion, i.e., at two distinct hubs. The underlying design problem partitions a given set of nodes into hubs and terminals, chooses a set of connections between the hubs such that the resulting backbone network is two-edge connected, and for each terminal chooses two hubs to provide the dual-homing backbone access. All of these decisions are jointly made based on some cost considerations. We give alternative formulations using cut inequalities, compare these formulations, provide a polyhedral analysis of the smallsized formulation, describe valid inequalities, study the associated separation problems, and design variable fixing rules. All of these findings are then utilized in devising an efficient branch-and-cut algorithm to solve this network design problem.

7 citations


"A branch-and-cut algorithm for two-..." refers background in this paper

  • ...[15] propose a branch-and-cut algorithm for the problem where the backbone network is 2EC and each user node is connected directly to two distinct hub nodes....

    [...]

Posted Content
TL;DR: In this article, a new network design model combining ring and tree structures under capacity constraints is proposed, which is based on ring trees which are the union of trees and 1-trees.
Abstract: We study a new network design model combining ring and tree structures under capacity constraints. The solution topology of this capacitated ring tree problem (CRTP) is based on ring trees which are the union of trees and 1-trees. The objective is the minimization of edge costs but could also incorporate other types of measures. This overall problem generalizes prominent capacitated vehicle routing and Steiner tree problem variants. Two customer types have to be connected to a distributor ensuring single and double node connectivity, respectively, while installing optional Steiner nodes. The number of ring trees and the number of customers supplied by such a single structure are bounded. After embedding this combinatorial optimization model in existing network design concepts, we develop a mathematical formulation and introduce several valid inequalities for the CRTP that are separated in our exact algorithm. Additionally, we use local search techniques to tighten the obtained upper bounds. For a set of literature-derived instances we consider various reliability scenarios and present computational results.

1 citations

Frequently Asked Questions (1)
Q1. What are the contributions in "A branch-and-cut algorithm for two-level survivable network design problems" ?

This paper approaches the problem of designing a two-level network protected against single-edge failures. The authors consider two survivable structures in both network levels. The aim of the problem is to minimize the total cost. The authors give integer programming formulations and valid inequalities for the different versions of the problem, solve them using a branch-and-cut algorithm, and discuss computational results.