# Energy and Environment-Aware Path Planning in Wireless Sensor Networks with Mobile Sink
## 4 Formulation for Optimal Solution
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\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Given Parameters**} \\ \hline
**Notation** & **Description** \\ \hline \(UM_{s}^{t}\) & The set of all urgent messages collected from the members of the cluster head s at time t, \\  & \(\forall s\in CHU^{t},CHU^{t}\in CH^{t},CH^{t}\in V,and\ t\in T\). \\ \(p\) & The mobile sink moving path, S at time t, which consists of a set of all vertices from \(v_{i}\) to \(v_{n}\), \\  & \(v_{i},v_{n}\in CH^{t},CH^{t}\in V,t\in T\). \\ C\({}_{ux}(t)\) & The movement cost from point \(u\) to \(x\), \(u,x\in CH^{t},CH^{t}\in V,t\in T\) \\ \(D_{ux}(t)\) & The movement delay from point \(u\) to \(x\), \(u,x\in CH^{t},CH^{t}\in V,t\in T\) \\ \(ZED_{i}^{t}\) & The zone \(i\)’s environmental data with environmental factor \(f\). \\ \(EI_{i}^{t}\) & Zone \(i\) environmental impact metric for single environmental factor \(f\). \\ \(MEI_{i}\) & Environmental impact metric of zone \(i\) for multiple environmental factors. \\ \(NEI_{i}\) & The neighboring environmental impact metric of zone \(i\). \\ \(FEI_{i}\) & The final environmental impact metric of the cluster head \(x\) in a zone \(i\) at time \(t\), \\ \(EIM_{x}^{t}(t)\) & The final environmental impact metric of the cluster head \(x\) in a zone \(i\) at time \(t\), \\ \(EI_{th}\) & The environmental impact threshold. \\ \(K_{ij}\) & The attenuation coefficient between zone \(i\) and \(j\). \\ \(ED_{ij}\) & the distance between zones i and j centers where \(i\
eq j\). \\ \(ED_{(x,MS)}(t)\) & The distance between both the cluster head x and sink node MS at time \(t\), \(x\in CH^{t},CH^{t}\in V,t\in T\). \\ \(H_{c(x,MS)}(t)\) & The hop count from the cluster head x to sink node MS at time \(t\), \(x\in CH^{t},CH^{t}\in V,t\in T\). \\ \(ME_{(u,x)}(t)\) & The energy needed to transfer mobile sink s from cluster head \(u\) to x at time \(t\), \\  & \(x\in CH^{t},CH^{t}\in V,t\in T\). \\ \(ZP_{i}(t)\) & The zone priority of zone \(i\) at time \(t\), \(i\in Z,t\in T\). \\ \(ZPM_{x}^{t}(t)\) & The zone priority metric of the cluster head x in zone \(i\) at time \(t\), \(x\in CH^{t},CH^{t}\in V,i\in Z,t\in T\). \\ _IDL_ & The initial deadline is defined at each source node by the application. \\ _deadline_ & The packet deadline is updated at each hop. \\ _Dd\({}_{x}\)_ & The required delivery delay for the message-based cluster head \(x\), \(x\in CHU^{t},CHU^{t}\in V\). \\ \(Ad_{x}\) & Cluster head average distance x, \(x\in CHU^{t},CHU^{t}\in V\). \\ \(ICD_{x}\) & Is the intra-cluster distance of node \(x\), \(x\in V\). \\ \(PRR_{xy}\) & The packet reception ratio for the link \((x,y)\), \((x,y)\in V\) and \(x\
eq y\). \\ \(PLR_{xy}\) & The packet loss ratio for the link \((x,y)\), \((x,y)\in V\) and \(x\
eq y\). \\ \(D_{xy}\) & The delay for the link \((x,y)\). \((x,y)\in V\) and \(x\
eq y\). \\ \(RE_{x}(t)\) & Residual energy of node x at time \(t\), \(x\in CH^{t},CH^{t}\in V\), \(t\in T\) \\ \(IE_{x}\) & Sensor node x initial energy, \(x\in CH^{t},CH^{t}\in V\) \\ \(HE_{xy}\) & Energy required for Single hop transmission from x to y, \(x,y\in CH^{t},CH^{t}\in V\) \\  & Energy consumption load function for each cluster head y at time \(t\), \\ \(ECL_{y}(t)\) & \(y\in NEB_{x},NE_{x}\in CH^{t},CH^{t}\in V,t\in T\) \\ \(NL_{y}(t)\) & The load of each cluster head \(x\) at time \(t\), \(y\in NEB_{x},NE_{x}\in CH^{t},CH^{t}\in V,t\in T\) \\ \(TM_{y}(t)\) & The total number of messages at cluster head \(y\), \(y\in CH^{t},CH^{t}\in V,t\in T\) \\ \(P_{y}^{t}\) & The set of all candidate paths between any pair \((y\), \(MS)\), \(\forall y\in CH^{t},CH^{t}\in V,and\ t\in T\) \\ \(NM_{x}\) & The neighbor set of node x that is not covered by any cluster head, \(x,NM_{x}\in V\). \\ \(NEB_{x}\) & The neighbor set of node x, \(x\), \(NEB_{x}\in V\). \\ \(CCHN_{x}\) & The cluster head neighbor set of cluster head x, \(x,CCHN_{x}\in CH^{t},CH^{t}\in V,and\ t\in T\). \\ \(FCHN_{x}\) & The final cluster head neighbor set of cluster head x, \(x,FCHN_{x}\in CH^{t},CH^{t}\in V,and\ t\in T\). \\ \hline \hline \end{tabular}
\begin{tabular}{c c} \hline \(\alpha_{y}^{p}\) & 1 if cluster head y is on path \(p\) to the mobile sink and 0 otherwise, \\  & \(\forall y\in FCHN_{x},p\in P_{y}^{t},FCHN_{x}\in CH^{t},CH^{t}\in V,and\ t\in T\). \\ \hline \multicolumn{2}{c}{Decision Variables} \\ \hline \(\bar{\xi}_{(i,i+1)}\) & 1 if the sink node \(MS\) moves from point i to i+1 and 0 otherwise, \(i,i+1\in CH^{t},CH^{t}\in V\). \\ \(X_{ux}^{t}\) & 1 if the mobile sink moves from cluster head u to cluster head x at time intervals \(t\) and 0 otherwise, \\  & \(\forall t\in T,u,\forall x,u\in CH^{t}\)_and_\(CH^{t}\in V\). \\ \(d_{xy}^{tu}\) & 1 if the cluster head x uses the link \((x,y)\) to transmit urgent message \(u\) through it to sink node and 0 otherwise, \(\forall u\in UM_{x}^{t},\forall s\in CHU^{t},\)\(x,y\in CH^{t},CHU^{t}\in V,and\ t\in T\). \\ \(\omega_{xy}^{su}\) & 1 if the cluster head x uses cluster head y to relay urgent message u of the source node s and 0 otherwise, \(\forall u\in UM_{x}^{t},\forall s\in CHU^{t},\)\(x,y\in CH^{t},CH^{t},CHU^{t}\in V,and\ t\in T\) \\ \hline \hline \end{tabular}
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