## 6 Mathematical Formulation Using Linear Programming
### Energy Consumption Minimization  
The imbalanced energy depletion among the network nodes degrade the network performance. In this regard, various routing algorithms are proposed to address this problem. Thus, energy minimization is performed based on objective function by following linear constraints. In both proposed schemes, energy consumption caused during transmission and reception of data packet. We formulate the objective function to optimize energy consumption (Equation (6)). \[Min\sum_{i=1}^{N}E_{consumed}(i)\ \ \ \ \forall\,i\,\epsilon\,N \tag{6}\]  
where \(E_{consumed}\) is the energy consumed per packet per node in the network. Initially, the energy depletion is because of packets transmission and reception which is counted as shown in Equation (7). \[E_{consumed}(ij)=\sum_{i=1}^{N}(E_{TX}+E_{RX}) \tag{7}\]  
In Equation (7), \(E_{consumed}\) between node \(i\) and node \(j\) is mainly due to the transmission of data \(E_{TX}\) over the distance \((D_{(ij)})\). The receiving energy \((E_{RX})\) depends on number of bits received in the data packet from sender node according to Equation (8). \[E_{TX}^{max}=P_{TX}\times(HS+L)/DR \tag{8}\]\[E_{RX}^{max}=P_{RX}\times(HS+L)/DR \tag{9}\]  
Equations (8) and (9) show optimal values of \(E_{TX}\) and \(E_{RX}\) and depend on transmission \(P_{TX}\) and receiving \(P_{RX}\) powers, respectively. Whereas, packet size is (\(HS+L\)) and DR depicts data rate. \[E_{total}=E_{initial}\times N \tag{10}\]  
\(E_{total}\) depicts the summation of network nodes energy as initial energy \((E_{initial})\) given in Equation (10). The \(E_{consumed}\) in each round (\(r\)) is stated in Equation (11) as,  
\[E_{consumed}=\sum_{r=1}^{r_{max}}(E_{consumed}(r)). \tag{11}\]  
For GDGOR-IA scheme, energy consumption due to depth adjustment of void nodes shown in Equation (12),  
\[E_{consumed}^{\prime}=\sum_{r=1}^{r_{max}}(E_{consumed}(r)+E_{DA}(r)), \tag{12}\]  
where \(E_{DA}(r)\) depicts the amount of energy dissipated in depth adjustment during each round which is added till maximum round \(r_{max}\) reaches. \[E_{DA}=N_{vm}\times(E_{DA}(n_{vm})). \tag{13}\]  
\(N_{vm}\) represents the number of void node. Objective function in Equation (6) is defined under following linear constraints:  
\[E_{(TX,RX)}\leq E_{i}^{initial}\quad\forall\,i\,\epsilon\,N \tag{14}\]  
\[D_{(s,d)}\leq R_{c}\quad\forall\,i\,\epsilon\,N \tag{15}\]  
\(D_{s,d}\) represents the distance between nodes \(s\) and \(d\) which must be less or equal to the \(R_{c}\) communication range. \[E_{consumed}(ij)=\sum_{i=1}^{N}(E_{TX}+E_{RX}) \tag{7}\]0  
In GRMC-SM, \(E_{consumed}\) is mainly due to single or multi-hop communication in the network. Therefore, \(E_{consumed}\) associated with this scheme can be computed by Equation (11). \[E_{consumed}(ij)=\sum_{i=1}^{N}(E_{TX}+E_{RX}) \tag{7}\]1  
The summation of transmission and reception energies \(E_{(TX,RX)}\) should remain less for successful transmission. While, \(E_{(TX,RX)}\) restricts receiving energy through Equation (18). \[E_{consumed}(ij)=\sum_{i=1}^{N}(E_{TX}+E_{RX}) \tag{7}\]2  
To limit the communication of the transmitter node within the transmission vicinity \(R_{TX}^{max}\), Equation (19) is used. Moreover, the distance should be greater than zero as given \(R_{TX}^{min}\) in Equation (20),  
\[E_{consumed}(ij)=\sum_{i=1}^{N}(E_{TX}+E_{RX}) \tag{7}\]3  
\[E_{consumed}(ij)=\sum_{i=1}^{N}(E_{TX}+E_{RX}) \tag{7}\]4  
\[E_{consumed}(ij)=\sum_{i=1}^{N}(E_{TX}+E_{RX}) \tag{7}\]5  
\[E_{consumed}(ij)=\sum_{i=1}^{N}(E_{TX}+E_{RX}) \tag{7}\]6  
**Graphical Analysis**: Let consider a scenario where 250 m be the transmission range and \(L_{s}\) levels i.e., \(L_{s}\) = [1; 2; 3; 4; 5]. The intention to make transmission \(L_{s}\) is to note down the pattern of energy dissipation based on \(L_{s}\) expressed in Equations (21) and (22).