# Lifetime Maximization via Hole Alleviation in IoT Enabling Heterogeneous Wireless Sensor Networks
## 3 LiMha
### Energy Minimization  
To portray reduction in the energy consumption once the data traffic load is estimated and distributed accordingly. We have carried out an optimization approach through which a feasible solution is computed. If the energy consumption satisfies the constraints then it is justified that data traffic is balanced. To prove it mathematically, linear programming is adopted. Linear programming is a mathematical approach which is broadly taken in to consideration while optimizing desired objective(s). The optimization process could be maximizing or minimizing depending upon the nature of objective function such as energy; it always needs to be minimized. An objective function must operate within the given linear constraints. We define the following objective function for the achieving optimal result in term of energy consumption. \[minimize\sum_{r=1}^{r_{max}}E_{tax}(r)\quad\forall r\in r_{max} \tag{12}\]  
Energy dissipation mainly includes the reception and transmission energies which are collectively known as energy tax \((E_{tax})\). The energy tax increases as the number of hops or number nodes involve in data packet delivering increases. If we want to save the node battery power, one of the factors must need to be analyzed. Here, we have minimized the number of nodes by exploiting divided and conquer mechanism. Due to the division of the network field; nodes are restricted only to the specific sector. Now, Equations (7) and (8) are combined to find out the total energy dissipated in one round. Equation (13) is formulated to portray the overall amount of energy dissipated in delivering data packets successfully at the destination. \[E_{tax}(r)=E_{tx}+\sum_{n=0}^{N}(E_{tx}+E_{rcv})\times(n\times P_{kt_{sink}}) \quad\forall n\in N \tag{13}\]  
\(E_{tx}\) is separately added because nodes in sink's range will forward data directly. On the other hand, second expression with summation is included to show both energies from upstream nodes toward the downstream nodes in the network field. \(n\) is zero because it is possible that due to death of nodes, few sectors could be empty. Now, to restrict our objective function, following constraints are defined. **Constraints:**  
\[Con_{1}: E_{tx},E_{rcv}\leq\varepsilon_{o} \forall n\in N \tag{14}\] \[Con_{2}: E_{F}\leq E_{F}^{min} \forall n\in N\] (15) \[Con_{5}: TR_{n}\leq TR_{n}^{max} \forall n\in N \tag{16}\]  
The purpose of Equation (14) is to ensure that energy required for transmission and reception should be less than the initial energy \(\varepsilon_{o}\) of the node. Similarly, \(Con_{2}\) shows the constraint regarding the selection of forwarder data, that a node having less energy consumption as compared to near by nodes will be picked as a forwarder in order to optimize the network lifetime. For receiving good quality signal, it should be transmitted within the communication range which is exactly forced by Equation (16). **Graphical Analysis:** The graphical analysis is presented because of its nature to provide clear visualization about precise indications of all possibilities within the bounded region. For two dimension scenarios, graphical approach is best fit to demonstrate optimal available options. The graphical analysis of our energy minimization is depicted in Figure 3.  
Let assume \(\varepsilon_{fs}=20\) nJ, \(\varepsilon_{amp}=20\) nJ, \(E_{eclec}=50\) nJ, \(d=60\) m, \(r=120\) m, \(\rho=10\), \(\zeta=10\), \(x=5\) m, \(\theta=45^{\circ}\), \(R=500\) m, \(N=150\). Following values are extracted via using aforementioned equations. \[0.037\leq E_{tx}\leq 0.186 \tag{17}\]  
\[0.00001285\leq E_{rcv}\leq 0.04 \tag{18}\]  
\[0.037\leq E_{tx}+E_{rcv}\leq 0.226 \tag{19}\]  
The enclosed region is calculated by keeping in mind the constraints given in Equations (14)-(16). Equations (17)-(19) are drawn to find out feasible region. An optimal solution is validated by each vertex of the feasible region. The solution can be validated from the points on the boundary of the feasible region.