## 4 Formulation for Optimal Solution
\begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{2}{l}{**Given Parameters**} \\ \hline
**Notation** & **Description** \\ \hline \(V\) & The monitored field’s sensor nodes. \\ \(S^{t}\) & Is the set of source nodes at time \(t\), \(t\in T\). \\ \(T\) & The time horizon. \\ \(Z\) & All zones \\ \(SN^{t}\) & Is the set of sink nodes deployed in the monitored field at time \(t\), \(t\in T\). \\ \(FSN^{t}\) & Final set of sink nodes that have an environmental impact metric larger than a threshold Value in the monitored field at time \(t\), \(t\in T\), \(FSN^{t}\in SN^{t}\). \\ \(M^{t}_{s}\) & Set of all messages collected from the source node \(s\) at time \(t\), \\ \(\forall s\in S^{t}\), \(S^{t}\in V\), and \(t\in T\). \(\forall s\in S\) \\ \(ZED^{f}_{i}\) & Zone \(i\)’s environmental data for factor \(f\). \\ \(EI^{f}_{i}\) & Zone \(i\)’s single environmental factor \(f\) impact metric. \\ \(MEI_{i}\) & Zone \(i\)’s multiple environmental impact metric. \\ \(NEI_{i}\) & Zone \(i\)’s neighboring environmental impact metric \\ \(FEI_{i}\) & Zone \(i\)’s final environmental impact metric. \\ \(EIM^{t}_{x}(t)\) & The sink/sensor node \(x\) final environmental impact metric in zone \(i\) at time \(t\), \(i\
eq j\). \\ \(EI_{th}\) & The environmental impact threshold. \\ \(K_{ij}\) & The coefficient of attenuation between zones \(i\) and \(j\). \\ \(ED_{ij}\) & Euclidean Distance (ED) between the center of zone \(i\) and \(j\), \(i\
eq j\). \\ \(ED_{(x,s_{j})}(t)\) & The Euclidean distance (ED) between the sensor node \(x\) and sink node \(s_{i}\) at time \(t\), \(i\
eq j\). \\ \(IDL\) & The initial deadline defined by the application at each source node. \\ \( deadline\) & Each hop’s packet deadline. \\ \(p^{s_{j}}_{y}\) & The set of all candidate paths between any pair \((y,s_{j})\), \(\forall s\in S\) \\ \(D^{s}_{s_{i}}\) & The desired message delivery speed from source s to sink \\ \(PRR_{xy}\) & Link’s (\(x\),\(y\)), packet reception ratio, \(i,j\in L\), and \(i\
eq j\). \\ \(PLR_{xy}\) & The packet loss ratio for the link (\(x\),\(y\)), \(i,j\in L\), and \(i\
eq j\). \\ \(D_{xy}\) & Link (\(x\),\(y\)) delay. \(i,j\in L\), and \(i\
eq j\). \\ \(RE_{x}(t)\) & Node \(x\)’s residual energy at time \(t\), \(i\in S\cup R\). \(i\in S\cup R\). \\ \(IE_{x}\) & Sensor node \(x\), initial energy, \(x\in V\)\(i\in S\cup R\). \(i\in S\cup R\). \\ \(HE_{xy}\) & Transmission energy from \(x\) to \(y\) in a single hop, \((i,j)\in L\) \\ \(ECL_{y}(t)\) & Energy load function for each sensor node \(y\) at time \(t\), \\ \(y\in NEB_{x}\), \(NEB_{x}\in V\), \(t\in T\) \\ \(NL_{y}(t)\) & Each sensor node \(x\) load at time \(t\), \(y\in NEB_{x}\), \(NEB_{x}\in V\), \(t\in T\) \\ \(TM_{y}(t)\) & Sensor node \(y\), total number of messages, \(y\in V\), \(t\in T\), \(i\in S\cup R-\{sink\}\) \\ \(SLM_{s_{i}}(t)\) & The sink load metric of each sink node \(s_{i}\), \(s_{i}\in SN^{t}\), \(t\in T\) \\ \(DhM^{s}_{s_{i}}\) & The distance hop metric between sink node \(s_{i}\) and each source node \(s\), \(\forall s\in S\) \\ \(mhc^{s_{i}}_{x}\) & The minimum hop count from any sensor node \(x\) to each sink node \(s_{i}\), \\ \(\forall x\in V\), \(\forall s_{i}\in SN^{t}\), \(t\in T\). \(i\), \(NEB_{i}\in S\cup R\) \\ \(SNE_{s_{i}}\) & sink node \(s_{i}\), neighbor set, \(s_{i}\in SN^{t}\). \(i\), \(NEB_{i}\in S\cup R\) \\ \(CNEN_{x}\) & sensor node \(x\), neighbor set, \(x\), \(CNEN_{x}\in V\). \\ \(FNEN_{x}\) & The final neighbor set of sensor node \(x\), \(x\), \(FNEN_{x}\in V\). \\ \hline \hline \multicolumn{2}{l}{**Indicator Parameter**} \\ \(a^{p}_{(y,s_{i})}\) & \(1\) if sensor node \(y\) is on path \(p\) to the chosen sink node \(s_{i}\) and \(0\) otherwise, \\ \(\forall y\in FNEN_{x},p\in P^{s}_{y}\), \(FNEN_{x}\in V\), and \(t\in T\). \\ \hline \multicolumn{2}{l}{**Decision Variables**} \\ \hline \(\xi^{sm}_{xy}\) & \(1\) if sensor node \(x\) utilizes the neighbor node \(y\) to relay message \(m\) of the source node s and \(0\) otherwise, \(\forall m\in M^{t}_{s}\), \(\forall s\in S^{t}\), \(x,y,S^{t}\in V\), and \(t\in T\) \\ \(\omega^{sm}_{xy}\) & \(1\) if the sensor node \(x\) utilizes neighbor node \(y\) to relay message \(m\) of the source node s and \(0\), otherwise, \(\forall m\in M^{t}_{s}\), \(\forall s\in S^{t}\), \(x,y,S^{t}\in V\), and \(t\in T\) \\ \hline \hline \end{tabular}
\end{table}