## 3 Classification of Optimization Objectives
### Multi-Objective Optimization Focussed on Design Related Problems in WSNs  
The design of WSNs is a relatively intricate task with significant influence on various performance parameters namely, quality, cost and efficiency of real life sensor applications. One of the design goals is to maximize the lifetime of the sensor network in a way that sensors effectively monitor the region of interest and communicate the observed information to the central processing station. [41]. A multi-objective optimization approach has been proposed in [42] for the modular design architecture of QoS aware routing protocol to ensure the homogeneous depletion rate of energy. A scheme for the minimization of energy consumption has been proposed in [43] by treating the design problem of beam pattern optimization as a multi-objective formulation. Mathematical formulations of some commonly used design related objectives have been depicted in Table 2. In the following we elaborate multi-objective optimization in sensor network related to various design problems.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Ref.** & **Objectives** & **Equation** & **Details** \\ \hline
[38] & Energy consumption, transmission radius, coverage area & \(\dfrac{A_{area}(N^{\prime})}{A_{s}}\) & Multi-objective function.Objective 1 is the coverage rate of sensor set \(N^{\prime}\).Objective 2 is financial cost of the sensor set \(N^{\prime}\) and Objective 3 is coverage energy consumption of the sensor set \(N^{\prime}\). \(A_{s}\) represents the total size of target area; \(A_{area}\) represents the monitoring size of the sensor set \(N^{\prime}\); \(r_{i}^{2}\) represents the sensing radius of node \(n_{i}\). \\ \hline
[42] & Customized QoS services for each traffic category & maximize & \(\dfrac{dist(i,s)-dist(j,s)}{dist(i,s)}\), \(\dfrac{e_{res(j)}}{e_{init}}\) (2) & Where \(dist(a,b)\) denotes geometric distance between nodes \(a\) and \(b\); \(e_{res(j)}\) denotes the residual energy level of the neighbor nodes \(j\) of \(i\); \(e_{init}\) denotes the initial energy level. \\ \hline
[45] & Energy consumption, system lifetime, coverage & \(\min f_{1}(P_{1},T,N)=\dfrac{1}{\eta}\sum_{i=1}^{N}(P_{ti}+\alpha_{ci})T_{i}\) & Multi-objective function. Objective 1: Optimization of energy consumption. Objective 2: optimization of system lifetime. Objective 3: coverage optimization problem. Objective 4: Optimization of the participating number of satellites. \(P_{ti}\) and \(T_{i}\) denotes optimal transmission power and transmission duration of node \(i\), respectively; \(\alpha_{ci}\) represents the equivalent circuit power consumption; \(N\) represents number of active satellite sensing nodes at any given time instant; \(\eta\) represents the efficiency of the power amplifier; \(\tau^{*}\) represents desired system lifetime; \(\tau_{sys}\) represents the system lifetime of a cluster networked system; \(\vartheta_{c}\) represents the coverage. \\ \hline
[49] & Energy consumption and spectrum sensing performance & \(\min F(X)=\big{(}C_{T},Q_{f},(-Q_{d})\big{)}^{T}\) & Where \(X=(\mu,\lambda 1,\lambda 2)\), where \(\lambda 1\) and \(\lambda 2\) are censoring thresholds, and \(\mu\) represents the probability that a node is turned off; \(C_{T}\) denotes the average energy consumption of the entire cognitive radio; \(Q_{D}\) denotes the global probability of spectrum sensing; \(Q_{f}\) denotes global probability of false alarm; \(T\) denotes samples during one sampling process. \\ \hline
[50] & Coverage preservation and energy conservation & \(F(x)=(f_{1}(x),......,f_{n}(x))^{T}\) & Where \(x\) is the decision variable vector. In general, \(f_{1},\dots,f_{n}\) are in conflict with each other, and then finding the optimum can be interpreted as finding a good trade-off between all \(f_{1},\dots,f_{n}\) of \(F\). \\ \hline \hline \end{tabular}
\end{table}
Table 2: Design related objectives in wireless sensor networks.