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Fuzzy particle swarm optimization algorithms for the open shortest path first weight setting problem

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Abstract

The open shortest path first (OSPF) routing protocol is a well-known approach for routing packets from a source node to a destination node. The protocol assigns weights (or costs) to the links of a network. These weights are used to determine the shortest paths between all sources to all destination nodes. Assignment of these weights to the links is classified as an NP-hard problem. The aim behind the solution to the OSPF weight setting problem is to obtain optimized routing paths to enhance the utilization of the network. This paper formulates the above problem as a multi-objective optimization problem. The optimization metrics are maximum utilization, number of congested links, and number of unused links. These metrics are conflicting in nature, which motivates the use of fuzzy logic to be employed as a tool to aggregate these metrics into a scalar cost function. This scalar cost function is then optimized using a fuzzy particle swarm optimization (FPSO) algorithm developed in this paper. A modified variant of the proposed PSO, namely, fuzzy evolutionary PSO (FEPSO), is also developed. FEPSO incorporates the characteristics of the simulated evolution heuristic into FPSO. Experimentation is done using 12 test cases reported in literature. These test cases consist of 50 and 100 nodes, with the number of arcs ranging from 148 to 503. Empirical results have been obtained and analyzed for different values of FPSO parameters. Results also suggest that FEPSO outperformed FPSO in terms of quality of solution by achieving improvements between 7 and 31 %. Furthermore, comparison of FEPSO with various other algorithms such as Pareto-dominance PSO, weighted aggregation PSO, NSGA-II, simulated evolution, and simulated annealing algorithms revealed that FEPSO performed better than all of them by achieving best results for two or all three objectives.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Pretoria, Pretoria, 0002, South Africa

    Mohammad Aijaz Mohiuddin, Salman A. Khan & Andries P. Engelbrecht

  2. Department of Computer Engineering, College of IT, University of Bahrain, Sakhir, Bahrain

    Salman A. Khan

Authors
  1. Mohammad Aijaz Mohiuddin
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  2. Salman A. Khan
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  3. Andries P. Engelbrecht
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Corresponding author

Correspondence to Mohammad Aijaz Mohiuddin.

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The authors declare that they have no conflict of interest. This article does not contain any studies with human participants or animals performed by any of the authors. Informed consent was obtained from all individual participants included in the study.

Appendices

Appendix A

1.1 Nomenclature

G :

Graph

N :

Set of nodes

n :

A single element in set N

A :

Set of arcs

A t :

Set of arcs representing shortest paths from all sources to destination node t

a :

A single element in set A. It can also be represented as (i,j)

s :

Source node

v :

Intermediate node

t :

Destination node

D :

Demand matrix

D :

[s,t] An element in the demand matrix that specifies the demand from source node s to destination node t; It can also be specified as d s t

w i j :

Weight on arc (i,j); if a=(i,j), then it can also be represented as w a

c i j :

Capacity on arc (i,j); if a=(i,j), then it can also be represented as c a

Φ:

Cost function

Φ i,j :

Cost associated with arc (i,j); if a=(i,j), then it can also be represented as Φ a

\({\delta _{u}^{t}}\) :

Outdegree of node u when destination node is t

δ +(u):

Outdegree of node u

δ −(u):

Indegree of node u

\({l_{a}^{t}}\) :

Load on arc a when destination node is t

l a :

Total traffic load on arc a

\(f^{(s,t)}_{a}\) :

Traffic flow from node s to t over arc a

S e t C A :

Set of congested arcs

Terminology

  1. 1.

    A single element in the set N is called a “Node”. It is represented as n.

  2. 2.

    A single element in the set A is called an “Arc” or “Link”. It is represented as a.

  3. 3.

    A set G=(N,A) is a graph defined as a finite nonempty set N of nodes and a collection A of pairs of distinct nodes from N.

  4. 4.

    A “directed graph” or “digraph” G=(N,A) is a finite nonempty set N of nodes and a collection A of ordered pairs of distinct nodes from N; each ordered pair of nodes in A is called a “directed arc”.

  5. 5.

    A digraph is “strongly connected” if for each pair of nodes i and j there is a directed path (i = n 1,n 2,...,n l = j) from i to j. A given graph G must be strongly connected for this problem.

  6. 6.

    A “demand matrix” is a matrix that specifies the traffic flow between s and t, for each pair (s,t)∈N×N.

  7. 7.

    (n 1,n 2,...,n l ) is a “directed walk” in a digraph G if (n i ,n i+1) is a directed arc in G for 1≤i≤l−1.

  8. 8.

    A “directed path” is a directed walk with no repeated nodes.

  9. 9.

    Given any directed path p=(i,j,k,...,l,m), the “length” of p is defined as w i j + w j k +... + w l m .

  10. 10.

    The “outdegree” of a node u is a set of arcs leaving node u i.e., {(u,v):(u,v)∈A}.

  11. 11.

    The “indegree” of a node u is a set of arcs entering node u i.e., {(v,u):(v,u)∈A}.

  12. 12.

    The input to the problem will be a graph G, a demand matrix D, and capacities of each arc.

  13. 13.

    The term MU refers to the maximum utilization. It is the highest load/capacity ratio of the network.

  14. 14.

    The term NOC refers to the number of congested links.

  15. 15.

    The term NUL refers to the number of unused links.

  16. 16.

    The term E refers to the total number of links in the network.

Appendix B

Tables 14 to 25 provide the quality of solutions obtained with respect to the associated swarm size for all test cases. Column 1 represents the number of particles in the swarm. Column 2 represents the average overall goodness using the UAO operator. Column 3 represents the percentage difference between the average overall goodness of the corresponding number of particles and the highest average overall goodness (given in asterisk) of the solutions. Note that the swarm size resulting in the highest average overall goodness is taken as the reference, and the difference for other swarm sizes is calculated with respect to the reference value. The differences were also statistically tested using Wilcoxon’s rank sum test.

Table 14 Effect of swarm size on overall cost for h100N280a with UAO
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Table 15 Effect of swarm size on overall cost for h100N360a with UAO
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Table 16 Effect of swarm size on overall cost for h50N148a with UAO
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Table 17 Effect of swarm size on overall cost for h50N212a with UAO
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Table 18 Effect of swarm size on overall cost for r100N403a with UAO
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Table 19 Effect of swarm size on overall cost for r100N503a with UAO
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Table 20 Effect of swarm size on overall cost for r50N228a with UAO
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Table 21 Effect of swarm size on overall cost for r50N245a with UAO
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Table 22 Effect of swarm size on overall cost for w100N391a with UAO
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Table 23 Effect of swarm size on overall cost for w100N476a with UAO
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Table 24 Effect of swarm size on overall cost for w50N169a with UAO
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Table 25 Effect of swarm size on overall cost for w50N230a with UAO
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Mohiuddin, M.A., Khan, S.A. & Engelbrecht, A.P. Fuzzy particle swarm optimization algorithms for the open shortest path first weight setting problem. Appl Intell 45, 598–621 (2016). https://doi.org/10.1007/s10489-016-0776-0

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