transmission range which is set to 20 in a square region which is
$50 m \times 50 m$.
\end{scriptsize}
-\caption{Illustration of a SN of size 10}\label{fig:sn}.
+\caption{Illustration of a Sensor Network of size 10}\label{fig:sn}.
\end{center}
\end{figure*}
The transmission consumed power of node $i$ is
$P_{ti} = c_l^s.y_l$ where $c_l^s$ is the transmission energy
consumption cost of link $l$, $l\in L$. This cost is defined
-as foolows: $c_l^s = \alpha +\beta.d_l^{n_p} $ wehre
+as foolows: $c_l^s = \alpha +\beta.d_l^{n_p} $ where
$d_l$ represents the distance of the link $l$,
$\alpha$, $\beta$, and $n_p$ are constant.
The reception consumed power of node $i$ is
The overall consumed power of the $i$ node is
$P_{si}+ P_{ti} + P_{ri}=
P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l +
-\sum_{l \in L} a_{il}^{-}.c^r.y_l \leq q.B_i.
-$
-
-The objective is thus to find $R$, $x$, $P_s$ which minimize
- $q$ under the following set of constraints
+\sum_{l \in L} a_{il}^{-}.c^r.y_l $.
+%\leq q.B_i.
+%$
+
+The objective is thus to find $R$, $x$, $P_s$ which maximizes
+the network lifetime $T_{\textit{net}}$, or equivalently which minimizes
+$q=1/{T_{\textit{net}}}$.
+Let $B_i$ is the initial energy in node $i$.
+One have the equivalent objective to find $R$, $x$, $P_s$ which minimizes
+$q^2$
+under the following set of constraints:
\begin{enumerate}
\item $\sum_{l \in L }a_{il}x_{hl} = \eta_{hi},\forall h \in V, \forall i \in N $
\item $ \sum_{h \in V}x_{hl} = y_l,\forall l \in L$
\item $\dfrac{\ln(\sigma^2/D_h)}{\gamma.P_{sh}^{2/3}} \leq R_h \forall h \in V$
\item \label{itm:q} $P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l +
-\sum_{l \in L} a_{il}^{-}.c^r.y_l \leq q.B_i, \forall i \in N$
+c^r.\sum_{l \in L} a_{il}^{-}.y_l \leq q.B_i, \forall i \in N$
+\item $\sum_{i \in N} a_{il}q_i = 0, \forall l \in L$
\item $x_{hl}\geq0, \forall h \in V, \forall l \in L$
\item $R_h \geq 0, \forall h \in V$
\item $P_{sh} > 0,\forall h \in V$
\begin{array}{l}
P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) \\
\qquad +
- \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q.B_i, \forall i \in N
+ \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q_i.B_i, \forall i \in N
\end{array}
$$
+and where the following constraint is added
+$$ $q_i > 0, \forall i \in N $$
+
They thus replace the objective of reducing
