## 3 The Proposed Model
### Design of WMAO Algorithm
Equations (26) and (27) demonstrate \(v_{(m)}^{dis}\) and \(v^{dis}(n)\), where \(Y_{z}\) indicates the normal node in \(z\)th cluster, \(G_{z}\) represents the CH of \(Z\)th cluster, the distance between the BS and CH is stated as \(G_{Z}-S_{s},\ G_{Z}-Y_{y}\) symbolizes the distance between the normal node and CH, \(Y_{z}-Y_{y}\) indicates the distance between two normal nodes, and \(M_{z}\) and \(M_{y}\vee\) represent the node amount that eliminates the \(Z\)th and \(y\)th clusters. \[v_{i}^{dis}=\frac{v_{(m)}^{dis}}{v_{(n)}^{dis}} \tag{25}\]

\[v_{(m)}^{dis}=\sum_{z=1}^{M_{z}}\biggl{[}\|G_{x}-S_{s}\|+\sum_{y=1}^{M_{y}}\| G_{Z}-Y_{y}\|\biggr{]} \tag{26}\]

\[v_{(n)}^{dis}=\sum_{z=1}^{M_{z}}\sum_{y=1}^{M_{y}}\|Y_{z}-Y_{y}\| \tag{27}\]

Equation (28) illustrates the FF of energy. The value \(v_{i}^{\prime\prime\prime\prime}\) is greater than 1 and the whole CH cumulative \(v_{(m)}^{ene}\) and \(v_{(n)}^{ene}\) concern the maximum energy value and the greater amount of CH. \[v_{i}^{\prime\prime\prime}=\frac{v_{(m)}^{\prime\prime\prime}}{v_{(n)}^{\prime \prime\prime}} \tag{28}\]Equation (29) shows the FF of delay \(v_{i}^{del}\) that ranges within \([0,\,1]\). \(v_{i}^{del}\) is directly proportional to each node inherent in the cluster. Consequently, the delay gets minimalized when the CH has a decreased number of nodes. The numerator indicates the high amount of CH and the denominator \(M_{M}\) indicates the total amount of nodes in the WSN. \[v_{i}^{del}=\frac{\max\left(\left\|G_{z}-Y_{z}\right\|\right)_{z=1}^{M_{c}}}{M _{M}} \tag{29}\]

## 4 Results and Discussion

This section inspects the routing performance of the WMAO-EAR method on WSNs. The presented WMAO-EAR model is tested under two node counts (NC) of 100 and 300. Table 1 provides a lifetime inspection of the WMAO-EAR model with existing methods in terms of FND, HND, and LND [24]. Figure 3 exhibits a brief lifetime assessment of the WMAO-EAR method under NC of 100. The results implied that the WMAO-EAR model attained an extended lifetime. For example, concerning FND, the WMAO-EAR model offered an increased FND of 1500 rounds, whereas the LEACH, CE-EC, SEED, NEH-CP, and HCEHUC models reached reduced FND of 466, 1000, 990, 780, and 2563 rounds, respectively. In terms of HND, the WMAO-EAR technique presented an increased HND of 4750 rounds, whereas the LEACH, CE-EC, SEED, NEH-CP, and HCEHUC techniques reached a reduced HND of 531, 4328, 3269, 2251, and 4525 rounds, respectively. Finally, in terms of LND, the WMAO-EAR algorithm provided an increased LND of 5432 rounds, while the LEACH, CE-EC, SEED, NEH-CP, and HCEHUC techniques achieved reduced LND of 275, 5230, 5072, 4500, and 5140 rounds, respectively. Figure 4 displays the detailed lifetime assessment of the WMAO-EAR approach under NC of 300. The results denote the WMAO-EAR method obtained an extended lifetime. For example, in terms of FND, the WMAO-EAR approach has rendered an increased FND of 2786 rounds, whereas the LEACH, CE-EC, SEED, NEH-CP, and HCEHUC approaches attained reduced FND of 528, 1091, 1610, 1814, and 2651 rounds, respectively. For HND, the WMAO-EAR method and HCEHUC models reached reduced HND of 616, 2088, 3566, 4221, and 4562 rounds, respectively. Finally, in terms of LND, the WMAO-EAR approach gave an increased LND of 5387 rounds, where the LEACH, CE-EC, SEED, NEH-CP, and HCEHUC methods reached reduced LND of 715, 2736, 5072, 4327, and 5017 rounds, respectively.