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Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that the network should be two-connected. A two-connected network fulfills the requirement that there should be at least two paths with no links in common between all pairs of nodes. One form of design model is to prescribe that the network should be composed of connected rings of links. The network design problem is then to choose links from a given network, and compose them into a number of rings. A ring is reliable in the sense that there always exist two ways of sending traffic, clockwise or counter-clockwise, which means that a ring fulfills the two-connectivity requirement. There is often a number of requirements on a ring, such as a limited length and limited number of nodes connected to the ring. This means that a ring network will include a number of rings, and traffic between rings must be possible. The traffic between rings is usually made at certain nodes, called transit nodes. Therefore all rings should be connected to at least one of the transit nodes. We focus on the case where we have two transit nodes in the network.

Each possible ring is associated with a certain fixed cost, and all links in a certain ring are given the same capacity. Reserve capacity is allocated according to certain principles. The number of possible rings in a network is an exponential function of the number of nodes in the network, so for larger networks is it impossible to a priori generate all possible rings.

We describe the problem, and model it as a linear integer programming problem, where a set of rings are assumed to be known. The usage of rings, i.e., the allocation of demand to rings, is determined. In practice, too many rings can not be included in the model. Instead we must be able to generate useful rings. A Lagrangean relaxation of the model is formulated, and the dual solution is used in order to derive reduced costs which can be used to generate new better rings. The information generated describes only the physical structure of the ring, not the usage of it. The ring generation problem is a modified traveling salesman subtour problem, which is known to be difficult to solve. Therefore, we focus on heuristic solution methods for this problem.

We also presents a column generation approach where the problem is modeled as a set covering problem. Here, a column describes both the topology of the ring and the exact usage of it. A similar ring generation problem appears as a subproblem, in order to generate new rings.

All methods are computationally tested on both real life data and randomly generated data, similar to real life problems.

##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 829
Mathematics
##### Identifiers
Local ID: 1563ISBN: 91-7373-668-6 (print)OAI: oai:DiVA.org:liu-22357DiVA: diva2:242670
##### Opponent
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-01-14
##### List of papers
1. A ring network design problem and heuristics for generating a set of feasible rings
Open this publication in new window or tab >>A ring network design problem and heuristics for generating a set of feasible rings
##### Abstract [en]

We discuss the problem of designing a telecommunication network with the survivability requirement that the network should be composed of connected rings of links. The work design problem is then to choose links from a given network, and compose them into a number of rings. Furthermore, the rings should be connected at certain transit nodes. The traffic between rings may pass through other rings. Each ring is associated with a certain fixed cost depending on the length of the ring. We describe the problem, modeled as a linear integer programming problem. We find a feasible solution to the problem by first find good rings in the network using two heuristics, and then solve the optimization model using only these rings. Finally, we give some computational results for different networks.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 16
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-22367 (URN)1575 (Local ID)1575 (Archive number)1575 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-08-29
2. Lagrangean price directive ring generation for network design
Open this publication in new window or tab >>Lagrangean price directive ring generation for network design
##### Abstract [en]

This paper addresses the problem of designing a telecommunication network with certain survivability requirements, namely that the network should be made up between connected rigs. This way single link failures do not kill the connection between any two nodes. One can make the network two-node-connected by including two specific nodes in all rings. This gives rise to a network design optimization problem with fixed costs on rings. In this paper we describe a solution approach for such problems, based on generation of rings. The approach is in principle a column generation technique, where the dual prices used for pricing out columns are obtained with the help of Lagrange duality, instead of the usual LP-duality. Computational tests are reported.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 17
##### Keyword
network design, rings, column generation, Lagrange multipliers
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-22370 (URN)1578 (Local ID)1578 (Archive number)1578 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-08-29
3. Calculating cost coefficients for generation of rings in network design
Open this publication in new window or tab >>Calculating cost coefficients for generation of rings in network design
##### Abstract [en]

We discuss a telecommunication network problem where the aim is to design a network that should be composed of connected rings of links. Each possible ring is associated with a certain fixed cost. The traffic between rings may pass through other rings, where the switch between two rings must be done at certain transit nodes. Each ring must pass at least one transit node. We describe the problem, modeled as a linear integer programming problem. We focus on calculating cost coefficients for ring generation using Lagrangean relaxation.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 18
##### Keyword
network design, rings, integer programming, column generation, lagrangean relaxation
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-22368 (URN)1576 (Local ID)1576 (Archive number)1576 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-08-29
4. A ring generation problem based on the traveling salesman subtour problem
Open this publication in new window or tab >>A ring generation problem based on the traveling salesman subtour problem
##### Abstract [en]

Survivability and high redundancy are two critical issues in field of telecommunications. If a telecommunication network is built up by rings, high redundancy can be established, since the traffic can be sent in either direction. Traffic is usually sent using one direction, and if a failure occurs, the opposite direction is used. There is often a number of requirements on a ring, such as a limit on the number of connected nodes. This means that the network will include a number of rings, and traffic between rings must be possible. Therefore, a network must include a number of transit nodes, where it is possible to send traffic between the rings. We focus on the case where network includes two transit nodes and each ring must include at least one transit node. Since the number of rings is enormous one needs to generate rings.

This paper discusses how to generate new rings, given that each node has a reward for connecting the node to the ring. The problem that occurs is a modification of a traveling salesman subtour problem with a additional constraint on the number of nodes connected. A problem formulation is given and some solution approaches are suggested. Two different scenarios are discussed, one where the aim is to modify an already existing ring, and one where the aim is to build a complete new ring. Some computational results are given for a real data network.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 19
##### Keyword
traveling salesman subtour problem, orienteering problem, prize collecting travelling salesman problem, ring generation
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-22369 (URN)1577 (Local ID)1577 (Archive number)1577 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-08-29
5. A ring network design problem solved by a ring generation and allocation approach
Open this publication in new window or tab >>A ring network design problem solved by a ring generation and allocation approach
##### Abstract [en]

The development of optical fibers in telecommunications has lead large changes in the field. When design a telecommunication network, capacity nowadays is cheap, and the minimal cost design tends to be a tree. Since such a design is very vulnerable for link or node failures, one often wish to include some kind of survivability requirement, for example that the network should be two-edge-connected or two-node-connected. Another form of design model is to prescribe that the network should be composed of connected rings of links. The network design problem is then to choose links from a give network, and compose them into a number of rings. Furthermore, the rings should be connected at certain transit nodes. Each possible ring is associated with a certain fixed cost, and all links in a certain ring are given the same capacity. Traffic between rings may pass through other rings, which is an important element of the problem. Finally, reserve capacity allocation according to certain principles is included. We describe the problem, modeled as a linear integer programming problem, and discuss different formulations and different solution methods. As the problem is quite difficult, we focus on heuristic solution methods, including elements of column generation and Lagrangean relaxation.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 20
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-22366 (URN)1574 (Local ID)1574 (Archive number)1574 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-08-29
6. A column generation approach for a ring network design problem
Open this publication in new window or tab >>A column generation approach for a ring network design problem
##### Abstract [en]

When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that there should be at least two paths between every pair of nodes in the network. A design model who fulfills this requirement is a network build up with rings. The network design problem is to choose links from a given network, and compose them into a number of rings. The rings are connected to each other at certain transit nodes. The number of possible rings is enormous, and each possible ring is associated with a certain fixed cost. A ring has a fixed capacity, however, we model it as a linear cost depending on the traffic using the ring and the length of the ring. We describe the problem, and model it is a set covering model, where a column describes how a specific ring is used. Even with a small set of rings, number of possible columns in the model is large. Therefore, a column generation approach is used to solve the set covering model with a given set of rings. An important part of the problem is to generate new rings, were the dual solution from the set covering model gives rewards on the nodes, representing a nodes’ wish to be included in a new ring. The ring generation problem is a modification of a traveling salesman subtour problem. New rings are generated using a heuristic. We present some computational results for a real data network and a number of random generated networks.

26 p.
##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 21
##### National Category
Engineering and Technology
##### Identifiers
urn:nbn:se:liu:diva-87193 (URN)
Available from: 2013-01-14 Created: 2013-01-14 Last updated: 2013-08-29

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Henningsson, Mathias

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Optimization The Institute of Technology
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