# Energy and Environment-Aware Path Planning in Wireless Sensor Networks with Mobile Sink
## 4 Formulation for Optimal Solution
### Optimal Path Planning Problem
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Given Parameters**} \\ \hline
**Notation** & **Description** \\ \hline \(r_{x}\) & 1 if the cluster head \(x\) has a final environmental metric greater than the threshold value and 0 otherwise, \(\forall x\in CH^{t},\text{ and }CH^{t}\in V\). \\ \(E_{n}\) & 1 if the difference between cluster head x’s environmental metric and cluster head n’s is less than zero and 0 otherwise,\(\forall x\in CH^{t},n\in CH^{t}-\{x\},\)\(i,j\in Z\) and \(CH^{t}\in V\). \\ \(g_{v}\) & 1 if the difference between the zone priority metric of cluster head \(x\)’s in zone \(i\), and cluster head \(v\)’s in zone \(j\) is less than 0, 0 otherwise.\(\forall x\in CH^{t},n\in CH^{t}-\{x\},\)\(i,j\in Z\) and \(CH^{t}\in V\). \\ \(h_{k}\) & 1 if the difference between the average distance metric value of cluster head x in zone \(i\) and cluster head \(k\) in zone \(j\) is less than zero and 0 otherwise, \(\forall x\in CH^{t},n\in CH^{t}-\{x\},\)\(i,j\in Z\) and \(CH^{t}\in V\). \\ \(I_{x}\) & 1 if the cluster head \(x\) has the highest environmental metric value compared to that of other cluster heads and 0 otherwise, \(\forall x\in CH^{t}\) and \(CH^{t}\in V\). \\ \(P_{x}\) & 1 if the cluster head \(x\) has the highest zone priority metric value compared to that of other cluster heads and 0 otherwise, \(\forall x\in CH^{t}\) and \(CH^{t}\in V\). \\ \(m_{x}\) & 1 if the cluster head \(x\) has the minimum average distance metric value compared to that of other cluster heads and 0 otherwise, \(\forall x\in CH^{t}\) and \(CH^{t}\in V\). \\ \(\psi_{x}\) & 1 if the node \(x\) is elected as a cluster head and 0 otherwise, \(x\in V\) \\ \(\delta^{t}_{x}\) & 1 if the node \(x\) is selected as a cluster head at time interval \(t\) and 0 otherwise, \(\forall t\in T,\forall x\in V\). \\ \(\epsilon_{f}\) & 1 if the difference between the quality metric values of node \(x\) and node \(f\) is less than zero and 0 otherwise, \(\forall x\in V,f\in V-\{x\}\) \\ \(\mu_{b}\) & 1 if the difference between the energy metric values node \(x\) and node \(b\) is less than zero and 0 otherwise, \(\forall x\in V,b\in V-\{x\}\) \\ \(q_{x}\) & 1 if the node \(x\) has the minimum quality metric value compared to that of other nodes and 0 otherwise, \(\forall x\in V\). \\ \(R_{x}\) & 1 if the node \(x\) has the highest energy metric value compared to that of other nodes and 0 otherwise, \(\forall x\in V\). \\ \(c_{y}\) & 1 if the cluster head \(y\) has relaying delay less than or equal to the desired delay and 0 otherwise, \(\forall y\in CCHN_{x},CCHN_{x}\in CH^{t},CH^{t}\in V,and\ t\in T\). \\ \(o_{y}\) & 1 if cluster head \(y\) can reach risk node and 0 otherwise, \(\forall y\in CCHN_{x},FCHN_{x}\in CH^{t},CH^{t}\in V,and\ t\in T\). \\ \(de_{y}\) & 1 if the cluster head \(y\) has the minimum value of the relaying delay compared with other cluster heads and 0 otherwise, \(\forall y\in FCHN_{x},FCHN_{x}\in CH^{t},CH^{t}\in V,and\ t\in T\). \\ \(a_{1}\) & 1 if the difference between the relaying delay value of cluster heads l and \(y\) is less than zero and 0 otherwise, \(\forall y\in FCHN_{x},I\in FCHN_{x}-\{y\},FCHN_{x}\in CH^{t},CH^{t}\in V,t\in T\). \\ \(e_{y}\) & 1 if the cluster head \(y\) has the highest value obtained from energy load function value compared to that of other cluster heads and 0 otherwise, \(\forall y\in FCHN_{x},FCHN_{x}\in CH^{t},CH^{t}\in V,and\ t\in T\). \\ \(b_{v}\) & 1 if the difference between the obtained values from energy load function of cluster head \(y\) and \(\
u\) is less than zero and 0 otherwise, \(\forall y\in FCHN_{x},v\in FCHN_{x}-\{y\},FCHN_{x}\in CH^{t},CH^{t}\in V,t\in T\). \\ \(v_{n}\) & 1 if the difference between the environmental metric value of cluster head y and n is less than zero and 0 otherwise, \(\forall y\in FCHN_{x},n\in FCHN_{x}-\{y\},FCHN_{x}\in CH^{t},CH^{t}\in V,t\in T\). \\ \(N_{y}\) & 1 if the cluster head \(y\) has the maximum environmental metric value compared to that of other cluster heads and 0 otherwise, \(\forall y\in FCHN_{x},FCHN_{x}\in CH^{t},CH^{t}\in V,t\in T\). \\ \hline \hline \end{tabular}
\end{table}