# A Comparative Evaluation of Nature Inspired Algorithms for Telecommunication Network Design

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

- $\mathcal{N}$
- set of all nodes;
- $\mathcal{T}$
- set of transponders;
- $\mathcal{S}$
- set of frequency slices;
- $\mathcal{E}$
- set of edges;
- ${\mathcal{P}}_{(n,{n}^{\prime},t)}$
- set of paths between nodes $n,{n}^{\prime}\in \mathcal{N}$ for transponder $t\in \mathcal{T}$; $p\subseteq \mathcal{E}$;
- $\mathcal{B}$
- set of bands;
- ${\mathcal{S}}_{b}$
- set of frequency slices used by band $b\in \mathcal{B}$; ${\mathcal{S}}_{b}\subseteq \mathcal{S}$; $\bigcup _{b\in \mathcal{B}}}{\mathcal{S}}_{b}=\mathcal{S$;
- ${\mathcal{S}}_{t}$
- set of frequency slices that can be used as starting; slices for transponder $t\in \mathcal{T}$; ${\mathcal{S}}_{t}\subseteq \mathcal{S}$.

- ${y}_{be}$
- binary variable, equal to 1 if band b is used on edge e and 0 otherwise;
- ${x}_{tn{n}^{\prime}ps}$
- binary variable that equals 1 if transponders t are installed between node n and node ${n}^{\prime}$, routed on path p and start on frequency slice $s\in {\mathcal{S}}_{t}$, and 0 otherwise.

- $\xi \left(b\right)$
- cost of using band b at a single edge;
- $\xi (t,b)$
- cost of using a pair of transponders t in band b;
- $v\left(t\right)$
- bitrate provided by transponder t;
- $d(n,{n}^{\prime})$
- bitrate demanded from node n to node ${n}^{\prime}$;
- $u(t,s,{s}^{\prime})$
- binary constant that equals 1 if transponder t using bandwidth starting at frequency slice s also uses frequency slice ${s}^{\prime}$; 0 otherwise.

## 3. Methodology

#### 3.1. Solution Model

- For all node pairs in set $\mathcal{N}$, a set of transponders $\mathcal{T}$ must be selected that are capable of meeting the demands for the node pair. The set is a 4-element vector $\mathcal{T}=\{n\xb7{t}_{1},n\xb7{t}_{2},n\xb7{t}_{3},n\xb7{t}_{4}\}$, where the first element is the number of 40G transponders used, the second element is the number of 100 G transponders used, the third element is the number of 200 G transponders used and fourth element is the number of 400 G transponders used. For example, $\mathcal{T}=\{2,1,0,3\}$ means that this set consists of two 40 G transponders, one 100 G transponder and three 400 Gb transponders. The set of transponders is chosen from four precalculated solutions, generated using linear programming.
- For each selected transponder a path should be determined, through which it will transmit data. The path $p\in \mathcal{P}$ is selected randomly by drawing one of the k-element set of predefined paths.
- Choosinga path is necessary to be able to assign each transponder $t\in \mathcal{T}$ a unique piece of bandwidth $s\in \mathcal{S}$ for data transmission. This is done using a heuristic RMLSA (routing modulation level and spectrum allocation) algorithm [14] that sequentially handles each demand and uses longest path first (LPF) sorting.

#### 3.2. Operators

## 4. Methods

#### 4.1. The Bees Algorithm

Algorithm 1: The bees algorithm. |

#### 4.2. Harmony Search

- For each node pair in set $\mathcal{N}$ choose solution which satisfies the demand.
- With probability ${p}_{a}$, a solution is chosen randomly from memory. There is also probability ${p}_{m}$ that the chosen solution will be modified using change path operator.
- Otherwise, a solution is created using change realization operator.

Algorithm 2: Harmony search. |

#### 4.3. Evolutionary Algorithm

Algorithm 3: The evolutionary algorithm ($\mu +\lambda $)–EA. |

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Structure of a model. Each demand consists of k 3-element vectors; k may be different for different demands.

**Figure 11.**Comparison of the convergence for the Polish network of all considered methods: artificial bee colony (BA), evolutionary algorithm (EA), harmony search algorithm (HS) and stochastic hill climbing algorithm (HC). Each curve is an average of 50 independent runs.

**Figure 12.**Comparison of the convergence for the USA network of all considered methods: artificial bee colony (BA), evolutionary algorithm (EA), harmony search algorithm (HS) and stochastic hill climbing algorithm (HC). Each curve is an average of 50 independent runs.

**Figure 13.**Comparison of the convergence for the American network of all considered methods: artificial bee colony (BA), evolutionary algorithm (EA), harmony search algorithm (HS) and stochastic hill climbing algorithm (HC). Each curve is an average of 50 independent runs.

**Figure 14.**Comparison of the convergence for 50-node the German network of all considered methods: artificial bee colony (BA), evolutionary algorithm (EA), harmony search algorithm (HS) and stochastic hill climbing algorithm (HC). Each curve is an average of 50 independent runs.

**Figure 15.**Convergence curves (cost vs. iterations) for different combinations of $\mu $ and $\frac{\lambda}{\mu}$ settings. Each curve is an average of 50 independent runs. Values of the $\frac{\lambda}{\mu}$ coefficient are described in the legend of the figure. The results were computed for the Polish network.

**Figure 16.**Convergence curves (cost vs. iterations) for different combinations of $\mu $ and $\frac{\lambda}{\mu}$ settings. Zoom of Figure 15.

**Figure 17.**Convergence curves (cost vs. iterations) for different combinations of $\mu $ and $\frac{\lambda}{\mu}$ settings. Each curve is an average of 50 independent runs. Values of the $\frac{\lambda}{\mu}$ coefficient are described in the legend of the figure. The results were computed for the Polish network.

**Figure 18.**Convergence curves (cost vs. iterations) for different combinations of $\mu $ and $\frac{\lambda}{\mu}$ settings. Zoom of Figure 17.

Network | #Nodes | #Links | #Demands |
---|---|---|---|

Germany10 | 10 | 14 | 45 |

Polish | 12 | 18 | 66 |

Germany15 | 15 | 23 | 105 |

Germany20 | 20 | 29 | 190 |

USA | 26 | 42 | 325 |

American | 39 | 61 | 741 |

Germany | 50 | 88 | 1225 |

Set | Set Settings |
---|---|

$\mathcal{N}$ | in Table 1 |

$\mathcal{E}$ | in Table 1 |

$\mathcal{S}$ | 768 slots |

$\mathcal{B}$ | 2 bands |

$\mathcal{T}$ | 4 transponders (${t}_{1}=40$ G, ${t}_{2}=100$ G, ${t}_{3}=200$ G and ${t}_{4}=400$ G) |

${\mathcal{S}}_{t}$ | ${\mathcal{S}}_{1}=\{1\cdots 380\}\cup \{385\cdots 764\}$ ${\mathcal{S}}_{2}=\{1\cdots 380\}\cup \{385\cdots 762\}$ ${\mathcal{S}}_{3}=\{1\cdots 378\}\cup \{385\cdots 762\}$ ${\mathcal{S}}_{4}=\{1\cdots 376\}\cup \{385\cdots 760\}$ |

Constant | Constant Settings |

$d(n,{n}^{\prime})$[Gbps] | e.g., $d(1,2)=70$ |

$\xi \left(b\right)$ | $\xi \left(1\right)=1$, $\xi \left(2\right)=2$ |

$\xi (t,b)$ | $\xi (1,1)=2$, $\xi (2,1)=5$, $\xi (3,1)=7$, $\xi (4,1)=9$ $\xi (1,2)=2.4$, $\xi (2,2)=6$, $\xi (3,2)=8.4$, $\xi (4,2)=11.8$ |

German10 | |||||||
---|---|---|---|---|---|---|---|

Method | Solution | #Evaluation | Time [sec.] | ||||

Best | Mean | std. dev. | Mean | Mean | std. dev. | Gap [%] | |

BA | 152 | 152.63 | 0.48 | 69,992 | 6990.8 | 170.7 | – |

EA | 152 | 152.59 | 0.49 | 21,220 | 2170.0 | 51.0 | – |

HS | 152 | 152.96 | 0.19 | 5293 | 529.18 | 25.86 | – |

HC | 153 | 153.96 | 0.19 | 5226 | 522.18 | 25.86 | – |

IP | 152 | – | – | – | 461.12 | – | 0.00 |

Polish | |||||||
---|---|---|---|---|---|---|---|

Method | Solution | #Evaluation | Time [sec.] | ||||

Best | Mean | std. dev. | Mean | Mean | std. dev. | Gap [%] | |

BA | 1321 | 1321.0 | 0.0 | 108,000 | 7650.8 | 165.2 | – |

EA | 1321 | 1321.6 | 1.3 | 40,000 | 8562.1 | 275.3 | – |

HS | 1321 | 1330.1 | 5.9 | 7641 | 420.3 | 90.1 | – |

HC | 1321 | 1321.7 | 1.4 | 2421 | 410.8 | 80.8 | – |

IP | 1321 | – | – | – | 367.0 | – | 0.00 |

German15 | |||||||
---|---|---|---|---|---|---|---|

Method | Solution | #Evaluation | Time [sec.] | ||||

Best | Mean | std. dev. | Mean | Mean | std. dev. | Gap [%] | |

BA | 281 | 281.85 | 0.35 | 70,084 | 7008.0 | 170.7 | – |

EA | 281 | 281.4 | 0.49 | 27,220 | 2710.0 | 62.0 | – |

HS | 281 | 281.93 | 0.47 | 5862 | 582.37 | 105.5 | – |

HC | 282 | 282.9 | 0.32 | 5440 | 544.5 | 24.8 | – |

IP | 281 | – | – | – | 1547.10 | – | 0.00 |

German20 | |||||||
---|---|---|---|---|---|---|---|

Method | Solution | #Evaluation | Time [sec.] | ||||

Best | Mean | std. dev. | Mean | Mean | std. dev. | Gap [%] | |

BA | 500 | 502.1 | 0.21 | 55,612 | 5561.0 | 417.7 | – |

EA | 500 | 506.0 | 0.70 | 29,770 | 2870.0 | 20.7 | – |

HS | 501 | 503.7 | 0.4 | 5245 | 524.5 | 47.1 | – |

HC | 501 | 504.0 | 0.3 | 6255 | 625.3 | 562.0 | – |

IP | 500 | – | – | – | 17304.75 | – | 0.00 |

USA | |||||||
---|---|---|---|---|---|---|---|

Method | Solution | #Evaluation | Time [sec.] | ||||

Best | Mean | std. dev. | Mean | Mean | std. dev. | Gap [%] | |

BA | 1846 | 1851.6 | 8.6 | 108000 | 18113.2 | 187.4 | – |

EA | 1846 | 1872.6 | 10.2 | 40,000 | 19,006.0 | 514.4 | – |

HS | 1866 | 1884.8 | 10.3 | 9057 | 1696.3 | 230.5 | – |

HC | 1862 | 1886.4 | 11.9 | 4868 | 1661.5 | 279.7 | – |

HY | 1846 | 1864.4 | 8.8 | 21248 | 9717.8 | 455.0 | – |

IP | 2366 | – | – | – | $3\xb7{10}^{5}$ | – | 25.7 |

American | |||||||
---|---|---|---|---|---|---|---|

Method | Solution | #Evaluation | Time [sec.] | ||||

Best | Mean | std. dev. | Mean | Mean | std. dev. | Gap [%] | |

BA | 4658 | 4698.3 | 40.0 | 210,819 | 57,299.3 | 452.4 | – |

EA | 4685 | 4751.9 | 26.2 | 40,000 | 62,470.9 | 706.9 | – |

HS | 5047 | 5136.1 | 46.8 | 9205 | 4931.5 | 734.3 | – |

HC | 4900 | 5051.4 | 85.5 | 23,407 | 30,819.3 | 6884.5 | – |

IP | 6020 | – | – | – | $6\xb7{10}^{5}$ | – | 33.2 |

German50 | |||||||
---|---|---|---|---|---|---|---|

Method | Solution | #Evaluation | Time [sec.] | ||||

Best | Mean | std. dev. | Mean | Mean | std. dev. | gap [%] | |

BA | 2968 | 3043.8 | 37.4 | 108,000 | 42,981.3 | 484.8 | – |

EA | 2851 | 2852.0 | 1.0 | 40,000 | 52,929.6 | 750.6 | – |

HS | 2979 | 2993.9 | 7.7 | 8805 | 5783.3 | 915.8 | – |

HC | 3088 | 3127.4 | 22.3 | 2894 | 2525.3 | 1048.0 | – |

HY | 2917 | 2959.7 | 13.2 | 21321 | 28166.6 | 3980.9 | – |

IP | 3196 | – | – | – | $6\xb7{10}^{5}$ | – | 13.1 |

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**MDPI and ACS Style**

Kozdrowski, S.; Żotkiewicz, M.; Wnuk, K.; Sikorski, A.; Sujecki, S.
A Comparative Evaluation of Nature Inspired Algorithms for Telecommunication Network Design. *Appl. Sci.* **2020**, *10*, 6840.
https://doi.org/10.3390/app10196840

**AMA Style**

Kozdrowski S, Żotkiewicz M, Wnuk K, Sikorski A, Sujecki S.
A Comparative Evaluation of Nature Inspired Algorithms for Telecommunication Network Design. *Applied Sciences*. 2020; 10(19):6840.
https://doi.org/10.3390/app10196840

**Chicago/Turabian Style**

Kozdrowski, Stanisław, Mateusz Żotkiewicz, Kacper Wnuk, Arkadiusz Sikorski, and Sławomir Sujecki.
2020. "A Comparative Evaluation of Nature Inspired Algorithms for Telecommunication Network Design" *Applied Sciences* 10, no. 19: 6840.
https://doi.org/10.3390/app10196840