# A Nature-Inspired Approach to Energy-Efficient Relay Selection in Low-Power Wide-Area Networks (LPWAN)

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Relay Selection Algorithm

#### 3.1. Introduction

#### 3.1.1. Problem Formulation

#### Relay Node Selection with Constraints

- ${V}_{J}={U}_{H}\cup R$
- ${E}_{J}=\{(u,r):u\in {U}_{H}\wedge r\in R\wedge r\in {N}_{H}\left(u\right)\}$
- $\forall u\in {U}_{H},\forall r\in R{deg}_{J}\left(u\right)=de{g}_{J}\left(r\right)=1$
- The sum of edge weights in graph J is the result of maximization:$$max\sum _{(u,w)\in {E}_{H}}\eta (u,w){x}_{uw}$$$$\sum _{u\in {U}_{H}}{x}_{uw}\u2a7d1,\forall w\in {W}_{H}$$$$\sum _{w\in {W}_{H}}{x}_{uw}=1,\forall u\in {U}_{H}$$

#### 3.2. Energy Consumption Model

#### 3.3. Heuristic Function

- The heuristic function $\eta $ promotes edges connecting potential relay nodes with a high energy surplus and a low maintenance cost for weak nodes. The value of $\eta (u,w)$ increases as the numerator (energy surplus ${E}_{w}^{+}$) rises and the denominator (weak node maintenance cost ${E}_{R{X}_{u}}+{E}_{T{X}_{w}}$) decreases, enhancing the overall attractiveness of the relay node.
- The energy surplus ${E}_{w}^{+}$ is calculated based on the device’s battery level, a crucial feature for battery-powered IoT devices.
- The Spreading Factor, a vital transmission parameter in LoRa technology that influences battery consumption, is considered in the heuristic function $\eta $. The formula of $\eta $ (12) appropriately accounts for this factor regarding the energy used to receive a packet from a weak node (component ${E}_{R{X}_{u}}$ calculated on (11)) and retransmit this packet (component ${E}_{T{X}_{w}}$ calculated on (10)).

#### 3.4. Relay Selection Algorithm

Algorithm 1 ACO relay selection algorithm. Iterative approach finding a set of relays and their assignment to weak nodes. Takes a weighted bipartite graph $H=({U}_{H},{W}_{H},{E}_{H},\eta )$ as input, where ${U}_{H}$ is a set of given weak nodes, ${W}_{H}$ is a set of candidates for relays, ${E}_{H}$ is a set of edges, $\eta $ is the weight function of the edges, t is the number of iterations, and m is the number of ants. The procedure returns the best found assignment of relays to weak nodes. | ||

1: | function aco_relay_selection($H=({U}_{H},{W}_{H},{E}_{H},\eta )$, t, m) | |

2: | ${M}_{best}\leftarrow \varnothing $ | ▹ initialize result |

3: | ${L}_{best}\leftarrow -\infty $ | ▹ initialize result’s weight |

4: | for $i\leftarrow 1$ to t do | ▹ for i-th iteration |

5: | $paths\leftarrow \varnothing $ | |

6: | for $k\leftarrow 1$ to m do | ▹ for k-th ant |

7: | ${Avail}^{k}\left({U}_{H}\right)\left(i\right)\leftarrow {U}_{H}$ | ▹ initialize unvisited weak nodes |

8: | ${Avail}^{k}\left({W}_{H}\right)\left(i\right)\leftarrow {\left\{1\right\}}^{|{W}_{H}|}$ | ▹ initialize unvisited potential relays |

9: | ${M}^{k}\left(i\right)$, ${L}_{M}^{k}\left(i\right)$ = generate_ant_path(H, ${Avail}^{k}\left({U}_{H}\right)\left(i\right)$, ${Avail}^{k}\left({W}_{H}\right)\left(i\right)$) [Algorithm 2] | |

10: | $paths\leftarrow paths\cup \{{M}^{k}\left(i\right),{L}_{M}^{k}\left(i\right)\}$ | |

11: | if ${L}_{M}^{k}\left(i\right)>{L}_{best}$ then | |

12: | ${L}_{best}={L}_{M}^{k}\left(i\right)$ | |

13: | ${M}_{best}={M}^{k}\left(i\right)$ | ▹ update best solution |

14: | for $M\in paths$ do | |

15: | update pheromone decay based on (14) ▹ update pheromone decay | |

16: | return ${M}_{best}$ |

Algorithm 2 Procedure for ant path generation. Returns weak node–relay assignments and their respective weights as found by the k-th ant in the i-th iteration. | ||

1: | function generate_ant_path($H=({U}_{H},{W}_{H},{E}_{H},\eta )$, $Avail{\left({U}_{H}\right)}^{k}\left(i\right)$, $Avai{l}^{k}\left({W}_{H}\right)\left(i\right)$) | |

2: | ${M}^{k}\left(i\right)\leftarrow \varnothing $ | ▹ initialize ant’s path |

3: | ${L}_{M}^{k}\left(i\right)\leftarrow 0$ | ▹ initialize ant path’s weight |

4: | while $|Avail{\left({U}_{H}\right)}^{k}\left(i\right)|>0$ do | ▹ while unvisited weak nodes exist |

5: | select $u\in {Avail}^{k}\left({U}_{H}\right)\left(i\right)$ | ▹ select random weak node |

6: | select node $w\in {N}_{u}^{k}\left(i\right)$ based on ${p}_{uw}^{k}$ (13) | ▹ select relay node |

7: | ${M}^{k}\left(i\right)\leftarrow {M}^{k}\left(i\right)\cup \left\{(u,w)\right\}$ | ▹ add the assignment to the ant’s path |

8: | ${L}_{M}^{k}\left(i\right)={L}_{M}^{k}\left(i\right)+\eta (u,w)$ | ▹ update ant path’s weight |

9: | ${Avail}^{k}\left({U}_{H}\right)\left(i\right)\leftarrow {Avail}^{k}\left({U}_{H}\right)\left(i\right)\setminus \left\{u\right\}$ | ▹ update unvisited weak nodes |

10: | $Avai{l}^{k}\left({W}_{H}\right)\left(i\right)\left[w\right]=0$ | ▹ update available potential relays |

11: | return ${M}^{k}\left(i\right)$, ${L}_{M}^{k}\left(i\right)$ | ▹ return the path and its weight |

#### 3.5. Computational Complexity

## 4. Parameters Analysis

## 5. Performance Evaluation

#### 5.1. Simulation Model

- Spreading Factor: determines a packet’s ToA, range, data rate, and energy consumption. The simulation model considered devices operating on different SFs (from 7 to 12).
- Transmission Power: refers to the power used by a transmitter to send signals (from −137 dBm to 14 dBm).

#### 5.2. Simulation Scenarios

^{2}, while R(1500, 3) and R(1500, 5) feature around 160 nodes per 1 km

^{2}. Therefore, the topologies include areas with lower and higher node densities in space. Among the end nodes, percentages of 3% and 5% were randomly selected as weak nodes. Each network topology scenario included LoRa gateways selected through the procedure from [37]. This gateway selection method typically identifies more than one gateway for the considered network topology scenarios R(1500, 3) and R(1500, 5); on average, there are approximately six gateways (standard deviation 0.6), realizing a multi-gateway schema. Conversely, the algorithm typically identifies a single access point for smaller and denser topologies R(1000, 3), R(1000, 5).

#### 5.2.1. Experiment Scenarios

#### Extensive Case

#### Demonstrative Case

#### 5.2.2. Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Mekki, K.; Bajic, E.; Chaxel, F.; Meyer, F. A comparative study of LPWAN technologies for large-scale IoT deployment. ICT Express
**2019**, 5, 1–7. [Google Scholar] [CrossRef] - Ikpehai, A.; Adebisi, B.; Rabie, K.M.; Anoh, K.; Ande, R.E.; Hammoudeh, M.; Gacanin, H.; Mbanaso, U.M. Low-power wide area network technologies for Internet-of-Things: A comparative review. IEEE Internet Things J.
**2018**, 6, 2225–2240. [Google Scholar] [CrossRef] - Semtech Corporation. LoRa® Technology|Semtech. 2023. Available online: https://www.semtech.com/lora/what-is-lora (accessed on 3 April 2024).
- Vangelista, L. Frequency shift chirp modulation: The LoRa modulation. IEEE Signal Process. Lett.
**2017**, 24, 1818–1821. [Google Scholar] [CrossRef] - Alliance, L. LoraWAN Relay Specification TS011-1.0.0. 2022. Available online: https://resources.lora-alliance.org/technical-specifications/ts011-1-0-0-relay (accessed on 27 November 2023).
- He, Q.; Lan, T.; Li, J.; Yuan, X.; Hu, Y. A Wireless Relay Assisted LPWAN for Condition Monitoring of Converter Stations. In Proceedings of the 2021 IEEE 6th International Conference on Signal and Image Processing (ICSIP), Nanjing, China, 22–24 October 2021; pp. 898–902. [Google Scholar]
- Xu, W.; Cai, G.; Fang, Y.; Mumtaz, S.; Chen, G. Performance Analysis and Resource Allocation for a Relaying LoRa System Considering Random Nodal Distances. IEEE Trans. Commun.
**2022**, 70, 1638–1652. [Google Scholar] [CrossRef] - Corporation, S. SX1272/3/6/7/8: LoRa Modem Designer’s Guide AN1200.13. Available online: https://www.openhacks.com/uploadsproductos/loradesignguide_std.pdf (accessed on 6 August 2019).
- van Eijk, P. LoRa Basics™ Modem Relay: A Low-Cost Battery Powered Network Extender; Technical Report; Semtech Corporation: Camarillo, CA, USA, 2023. [Google Scholar]
- Sanfratello, A.; Mingozzi, E.; Marcelloni, F. Enabling Relay-Based Communication in LoRa Networks for the Internet of Things: Design Implementation and Experimental Evaluation. Master’s Thesis, University of Pisa, Pisa, Italy, 2016. [Google Scholar]
- Sisinni, E.; Ferrari, P.; Fernandes Carvalho, D.; Rinaldi, S.; Marco, P.; Flammini, A.; Depari, A. LoRaWAN Range Extender for Industrial IoT. IEEE Trans. Ind. Inform.
**2020**, 16, 5607–5616. [Google Scholar] [CrossRef] - Borkotoky, S.S.; Schilcher, U.; Bettstetter, C. Cooperative Relaying in LoRa Sensor Networks. In Proceedings of the 2019 IEEE Global Communications Conference (GLOBECOM), Big Island, HI, USA, 9–13 December 2019; pp. 1–5. [Google Scholar] [CrossRef]
- Barrachina-Muñoz, S.; Bellalta, B.; Adame, T.; Bel, A. Multi-hop communication in the uplink for LPWANs. Comput. Netw.
**2017**, 123, 153–168. [Google Scholar] [CrossRef] - Sisinni, E.; Carvalho, D.F.; Ferrari, P.; Flammini, A.; Silva, D.R.C.; Da Silva, I.M.D. Enhanced flexible LoRaWAN node for industrial IoT. In Proceedings of the 2018 14th IEEE International Workshop on Factory Communication Systems (WFCS), Imperia, Italy, 13–15 June 2018; pp. 1–4. [Google Scholar] [CrossRef]
- Xu, W.; Cai, G.; Fang, Y.; Chen, G. Performance Analysis of a Two-Hop Relaying LoRa System. In Proceedings of the 2021 IEEE/CIC International Conference on Communications in China (ICCC), Xiamen, China, 28–30 July 2021; pp. 540–545. [Google Scholar] [CrossRef]
- Peppas, K.; Chronopoulos, S.K.; Loukatos, D.; Arvanitis, K. New Results for the Error Rate Performance of LoRa Systems over Fading Channels. Sensors
**2022**, 22, 3350. [Google Scholar] [CrossRef] [PubMed] - Fu, J.; Ma, X.; Yu, H.; Dai, K. Distributed energy-efficient wireless sensing and information fusion via event-driven and state-rank activation. Wirel. Netw.
**2024**, 30, 1–15. [Google Scholar] [CrossRef] - Lee, S.; Lee, J.; Park, H.S.; Choi, J.K. A Novel Fair and Scalable Relay Control Scheme for Internet of Things in LoRa-based Low-Power Wide-Area Networks. IEEE Internet Things J.
**2020**, 8, 5985–6001. [Google Scholar] [CrossRef] - Mugerwa, D.; Nam, Y.; Choi, H.; Shin, Y.; Lee, E. SF-Partition-Based Clustering and Relaying Scheme for Resolving Near-Far Unfairness in IoT Multihop LoRa Networks. Sensors
**2022**, 22, 9332. [Google Scholar] [CrossRef] [PubMed] - Grochla, K.; Strzoda, A.; Marjasz, R.; Głomb, P.; Książek, K.; Łaskarzewski, Z. Energy-Aware Algorithm for Assignment of Relays in LP WAN. ACM Trans. Sens. Netw.
**2022**, 18, 1–23. [Google Scholar] [CrossRef] - Li, Y.; Liao, C.; Wang, Y.; Wang, C. Energy-Efficient Optimal Relay Selection in Cooperative Cellular Networks Based on Double Auction. IEEE Trans. Wirel. Commun.
**2015**, 14, 4093–4104. [Google Scholar] [CrossRef] - Ma, B.; Shah-Mansouri, H.; Wong, V.W.S. A matching approach for power efficient relay selection in full duplex D2D networks. In Proceedings of the 2016 IEEE International Conference on Communications (ICC), Kuala Lumpur, Malaysia, 22–27 May 2016; pp. 1–6. [Google Scholar] [CrossRef]
- Kuhn, H.W. The Hungarian method for the assignment problem. Nav. Res. Logist. Q.
**1955**, 2, 83–97. [Google Scholar] [CrossRef] - Ochoa, M.N.; Guizar, A.; Maman, M.; Duda, A. Evaluating LoRa energy efficiency for adaptive networks: From star to mesh topologies. In Proceedings of the 2017 IEEE 13th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob), Rome, Italy, 9–11 October 2017; pp. 1–8. [Google Scholar] [CrossRef]
- Cai, J.; Shen, X.; Mark, J.W.; Alfa, A.S. Semi-distributed user relaying algorithm for amplify-and-forward wireless relay networks. IEEE Trans. Wirel. Commun.
**2008**, 7, 1348–1357. [Google Scholar] [CrossRef] - Korte, B.H.; Vygen, J.; Korte, B.; Vygen, J. Combinatorial Optimization; Springer: Berlin/Heidelberg, Germany, 2011; Volume 1. [Google Scholar]
- Galil, Z. Efficient algorithms for finding maximum matching in graphs. ACM Comput. Surv. (CSUR)
**1986**, 18, 23–38. [Google Scholar] [CrossRef] - Dorigo, M.; Birattari, M.; Stutzle, T. Ant colony optimization. IEEE Comput. Intell. Mag.
**2006**, 1, 28–39. [Google Scholar] [CrossRef] - Casals, L.; Mir, B.; Vidal, R.; Gomez, C. Modeling the energy performance of LoRaWAN. Sensors
**2017**, 17, 2364. [Google Scholar] [CrossRef] [PubMed] - Raza, U.; Kulkarni, P.; Sooriyabandara, M. Low Power Wide Area Networks: An Overview. IEEE Commun. Surv. Tutor.
**2017**, 19, 855–873. [Google Scholar] [CrossRef] - LoRa Alliance. LoRaWAN Regional Parameters RP002-1.0.4. 2024. Available online: https://resources.lora-alliance.org/technical-specifications/rp002-1-0-4-regional-parameters (accessed on 2 February 2024).
- Gao, W.; Du, W.; Zhao, Z.; Min, G.; Singhal, M. Towards Energy-Fairness in LoRa Networks. In Proceedings of the 2019 IEEE 39th International Conference on Distributed Computing Systems (ICDCS), Dallas, TX, USA, 7–10 July 2019; pp. 788–798. [Google Scholar] [CrossRef]
- Cormen, T.H.; Leiserson, C.E.; Rivest, R.L.; Stein, C. Introduction to Algorithms; MIT press: Cambridge, MA, USA, 2022. [Google Scholar]
- Maximum Weight Matching. Available online: https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.matching.max_weight_matching.html#id1 (accessed on 9 January 2024).
- Jungnickel, D. Weighted matchings. In Graphs, Networks and Algorithms; Springer: Berlin/Heidelberg, Germany, 2008; pp. 419–456. [Google Scholar]
- Marjasz, R.; Grochla, K.; Strzoda, A.; Laskarzewski, Z. Simulation Analysis of Packet Delivery Probability in LoRa Networks. In Proceedings of the Computer Networks; Gaj, P., Sawicki, M., Kwiecień, A., Eds.; Springer International Publishing: Cham, Switzerland, 2019; pp. 86–98. [Google Scholar] [CrossRef]
- Frankiewicz, A.; Glos, A.; Grochla, K.; Łaskarzewski, Z.; Miszczak, J.; Połys, K.; Sadowski, P.; Strzoda, A. LP WAN gateway location selection using modified k-dominating set algorithm. In Proceedings of the Modelling, Analysis, and Simulation of Computer and Telecommunication Systems: 28th International Symposium, MASCOTS 2020, Nice, France, 17–19 November 2020; Revised Selected Papers 28. Springer: Cham, Switzerland, 2021; pp. 209–223. [Google Scholar]
- Ross, S.M. Introduction to Probability Models; Chapter The Unifrom Random Variable; Academic Press: Cambridge, MA, USA, 2014; pp. 31–32. [Google Scholar]

**Figure 1.**Edge weight function (12) distribution depending on SF parameters $S{F}_{uw}$ and $S{F}_{wg}$ for weak node $u\in {U}_{H}$ and relay candidate $w\in {W}_{H}$.

**Figure 2.**Distribution of ACO’s mean solution quality, depending on the problem’s size and method’s hyperparameters ($\alpha $ and $\beta $). Sparse and complete bipartite graphs are included in the results.

**Figure 3.**Convergence of the ACO algorithm for problems of varying sizes, specifically, the mean and standard deviations of the best solution at each iteration: (

**a**) convergence of ACO for a relatively small complete graph (graph density 100%) $|{U}_{H}|=|{W}_{H}|=100$ and (

**b**) convergence of ACO for a medium-sized sparse graph (graph density 10%) $|{U}_{H}|=|{W}_{H}|=1000$.

**Figure 4.**Change in device battery level consumption during the network’s operation over a 10-year period. The figures show the battery life for relay devices and other network end devices. (

**a**) ACO relay selection method; (

**b**) EK relay selection method; (

**c**) reference relay selection method [21]. In the figure, the depletion of the battery of one of the selected relay devices is visible. The curve at the bottom of the chart that dips below zero in the 50th month corresponds to a relay device that has run out of battery.

SF | Packet ToA [s] (${\mathit{T}}_{\mathit{packet}}$) | ${\mathit{E}}_{\mathbf{TX}}$ [mAs] | ${\mathit{E}}_{\mathbf{RX}}$ [mAs] |
---|---|---|---|

7 | 0.118 | 4.366 | 0.767 |

8 | 0.215 | 7.955 | 1.3975 |

9 | 0.39 | 14.43 | 2.535 |

10 | 0.698 | 25.826 | 4.537 |

11 | 1.56 | 57.72 | 10.14 |

12 | 2.796 | 103.452 | 18.174 |

Component | Cost and Multiplicity |
---|---|

line 7: ${Avail}^{k}\left({U}_{H}\right)\left(i\right)\leftarrow {U}_{H}$ | ${c}_{1}\xb7{n}_{1}$ |

line 8: ${Avail}^{k}\left({W}_{H}\right)\left(i\right)\leftarrow {\left\{1\right\}}^{|{W}_{H}|}$ | ${c}_{2}\xb7{n}_{2}$ |

line 9: Algorithm 2 | ${c}_{3}\xb7\sum _{j=1}^{{n}_{1}+1}j+{c}_{4}\xb7\sum _{j=1}^{{n}_{1}}(j+{n}_{2})$ (based on Table 3) |

line 15: update pheromone level based on [14] | ${c}_{5}\xb7{n}_{1}$ |

**Table 3.**Computational cost and multiplicity of the significant components in Algorithm 2 within a whole loop (4).

Component | Cost and Multiplicity |
---|---|

line 4: while $|Avail{\left({U}_{H}\right)}^{k}\left(i\right)|>0$ | ${c}_{3}\xb7\sum _{j=1}^{{n}_{1}+1}j$ |

line 6: choose vertex $w\in {N}_{u}^{k}\left(i\right)$ based on ${p}_{uw}^{k}$ [13] | $c}_{6}\xb7\sum _{j=1}^{{n}_{1}}{n}_{2$ |

line 9: ${Avail}^{k}\left({U}_{H}\right)\left(i\right)\leftarrow {Avail}^{k}\left({U}_{H}\right)\left(i\right)\setminus \left\{u\right\}$ | ${c}_{7}\xb7\sum _{j=1}^{{n}_{1}}j$ |

Method | Computational Complexity |
---|---|

ACO | $\mathcal{O}(t\xb7m\xb7{E}_{H})$ |

EK [27] ^{1} | $\mathcal{O}\left({V}_{H}^{3}\right)$ |

Reference [21] ^{2} | $\mathcal{O}\left({V}_{H}^{3}\right)$ |

**Table 5.**Average and standard deviations of the running time and accuracy of the heuristic ACO method versus the exact EK method for large sparse graphs.

${\mathit{U}}_{\mathit{H}}\times {\mathit{W}}_{\mathit{H}}$ | Graph Density | Avg. Time [s] ACO | Avg. Time [s] EK | ACO Avg. Time Savings | Avg. Accuracy ACO | Avg. Accuracy EK |
---|---|---|---|---|---|---|

${10}^{3}\times {10}^{4}$ | $5\%$ | 752 $(\pm 12)$ | 1782 $(\pm 267)$ | $58\%$ | $99\%$ $(\pm 0.0002)$ | $100\%$ |

$10\%$ | 1709 $(\pm 75)$ | 3889 $(\pm 818)$ | $56\%$ | $98\%$ $(\pm 0.01)$ | $100\%$ | |

${10}^{3}\times {10}^{5}$ | $5\%$ | 12,971 $(\pm 897)$ | 15,296 $(\pm 23)$ | $15\%$ | $99\%$ $(\pm 0.0009)$ | $100\%$ |

$10\%$ | 29,556 $(\pm 1997)$ | 41,935 $(\pm 9772)$ | $30\%$ | $97\%$ $(\pm 0.002)$ | $100\%$ |

Network Topology Scenario | Total Nbr. of Nodes | Area [m] | Weak Nodes |
---|---|---|---|

R(1000, 3) | 1000 | $1000\times 1500$ | $3\%$ |

R(1000, 5) | 1000 | $1000\times 1500$ | $5\%$ |

R(1500, 3) | 1500 | $2500\times 3750$ | $3\%$ |

R(1500, 5) | 1500 | $2500\times 3750$ | $5\%$ |

**Table 7.**Results of side-by-side runs of the proposed methods (ACO and EK) and reference [21] algorithm, specifically, the mean and standard deviations for the energy usage in the whole network with a set of relays selected by the three methods in each simulation scenario. The statistics were calculated for a population of 30 randomly generated network topologies.

Network Topology Scenario | Method | Mean Battery Usage (%) | Standard Deviation (%) |
---|---|---|---|

R(1000, 3) | EK | 5.9550 | 1.2871 |

ACO | 5.9550 | 1.2870 | |

Referential [21] | 5.9546 | 1.2868 | |

R(1000, 5) | EK | 6.2125 | 1.4176 |

ACO | 6.2125 | 1.4176 | |

Referential [21] | 6.2123 | 1.4179 | |

R(1500, 3) | EK | 25.7492 | 0.7899 |

ACO | 25.7480 | 0.7894 | |

Referential [21] | 25.7494 | 0.7898 | |

R(1500, 5) | EK | 25.4126 | 0.7799 |

ACO | 25.4122 | 0.7803 | |

Referential [21] | 25.4180 | 0.7755 |

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**MDPI and ACS Style**

Strzoda, A.; Grochla, K.
A Nature-Inspired Approach to Energy-Efficient Relay Selection in Low-Power Wide-Area Networks (LPWAN). *Sensors* **2024**, *24*, 3348.
https://doi.org/10.3390/s24113348

**AMA Style**

Strzoda A, Grochla K.
A Nature-Inspired Approach to Energy-Efficient Relay Selection in Low-Power Wide-Area Networks (LPWAN). *Sensors*. 2024; 24(11):3348.
https://doi.org/10.3390/s24113348

**Chicago/Turabian Style**

Strzoda, Anna, and Krzysztof Grochla.
2024. "A Nature-Inspired Approach to Energy-Efficient Relay Selection in Low-Power Wide-Area Networks (LPWAN)" *Sensors* 24, no. 11: 3348.
https://doi.org/10.3390/s24113348