# Entropy Weighted TOPSIS Based Cluster Head Selection in Wireless Sensor Networks under Uncertainty
## 5 Numerical Experiment and Discussions  
Unsafe or unreliable communication has yielded detrimental consequences, including increased noise and adverse effects on sensor node batteries. The battery life directly affects the network's longevity, which is closely tied to residual energy. Furthermore, the presence of noisy data necessitates extra caution and thorough processing. Lastly, the distance between sensors, logic, and actuators is critical in facilitating replacements. This incident represents the uncertainty of WSNs. For computational experiments, we used a thousand nodes of WSNs where all the nodes were randomly distributed within a \(100\times 100\,\mathrm{m}^{2}\) area and ran the simulation on NS2 for the first round. In our experiment, we utilized the OPNET modeler to assess the MAC layer functionality of 802.15.4 slotted CSMA/CA. Here, we also considered uncertain parameters for the entire network setup, which has been shown in Table 3. Also, we estimated the optimum range of \(O_{c}\). Here, we considered \(\mathrm{N}_{n}=100\) nodes, \(\mathrm{Z}=100\,\mathrm{m}\), \(e_{fs}=10\,\mathrm{pJ}\), \(e_{\mathrm{mp}}=0.0013\,\mathrm{pJ}\), and \(76\,\mathrm{m}<db<168\,\mathrm{m}\). Therefore, the expected optimum number of clusters was in the range (1, 10), i.e., \(1<O_{c}<10\). \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Cluster Head** & **Residual Energy** & \begin{tabular}{l} **Number of** \\ **Neighbors** \\ \end{tabular} & \
\begin{tabular}{l} **Distance from** \\ **the Sink** \\ \end{tabular} &
\begin{tabular}{l} **Average Distance of** \\ **Clusters Nodes** \\ \end{tabular} & **Distance Ratio** & **Reliability** \\ \hline CH1 & 0.9695 & 8 & 157.203 & 13.232 & 0.0819 & 0.92 \\ CH2 & 0.9654 & 4 & 77.223 & 15.527 & 0.0774 & 0.96 \\ CH3 & 0.9698 & 8 & 141.173 & 26.937 & 0.0442 & 0.92 \\ CH4 & 0.9653 & 7 & 135.059 & 31.049 & 0.0396 & 0.93 \\ CH5 & 0.9688 & 4 & 92.444 & 47.752 & 0.0318 & 0.96 \\ CH6 & 0.9641 & 3 & 115.069 & 22.688 & 0.0528 & 0.97 \\ CH7 & 0.9647 & 4 & 85.988 & 22.348 & 0.0564 & 0.96 \\ CH8 & 0.9657 & 5 & 106.367 & 24.433 & 0.0503 & 0.95 \\ CH9 & 0.9649 & 6 & 102.181 & 15.694 & 0.0735 & 0.94 \\ CH10 & 0.9656 & 10 & 93.391 & 33.724 & 0.0404 & 0.9 \\ CH11 & 0.9698 & 9 & 119.436 & 17.016 & 0.0671 & 0.91 \\ CH12 & 0.9688 & 6 & 109.224 & 28.863 & 0.0438 & 0.94 \\ CH13 & 0.9698 & 2 & 85.158 & 29.5 & 0.0456 & 0.98 \\ CH14 & 0.9656 & 10 & 147.868 & 17.706 & 0.0632 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Decision parameter for selecting cluster heads. It ought to be noted that the signed distance method mentioned in Section 2.2 has been used to transform fuzzy parameters into defuzzified values. For example, \(I_{i}=(0.7,1.0,1.2)\) and \(D(\widetilde{I}_{i},\widetilde{0})=0.25(0.7+2\times 1.0+1.2)=0.25\times 3.9=0.975\). Other parameters have undergone a similar computation. Using Equation (8), we have calculated the value of \(O_{c}\). It is to be noted that the value of \(O_{c}\) lies between \(1<O_{c}<10\), and consequently, we have chosen the value of \(O_{c}=9\) for the purpose of the numerical experiment. Hence, we have selected nine cluster heads based on six criteria: residual energy, number of neighbors, distance from the sink, average distance of cluster node, distance ratio, and reliability using the MCDM approach. Here, Table 3 is a decision matrix of our proposed problem. Using Table 3, we have calculated weight vectors using the entropy method described in Section 2.3. The weight vector \(w\) using Section 2.3 has been computed as \(w=(0.156,0.176,0.167,0.175,0.170,0.156)\). Further, we have calculated \(sm_{i}^{+}\) and \(sm_{i}^{-}\) using Formulas (2) and (3). These are shown in Table 4.  
Using separation measures of each alternative, we have calculated the closeness coefficient using Equation (4), which has been shown in Table 5.  
The alternative with the highest rank is regarded as the best choice. Therefore, we have chosen nine cluster heads based on the closeness coefficient.