-Second, we define several binary and integer variables. Hence, each binary
-variable $X_{k}$ determines the activation of sensor $k$ in the sensing phase
-($X_k=1$ if the sensor $k$ is active or 0 otherwise). $M^j_i$ is an integer
-variable which measures the undercoverage for the coverage interval $i$
-corresponding to sensor~$j$. In the same way, the overcoverage for the same
-coverage interval is given by the variable $V^j_i$.
-
-If we decide to sustain a level of coverage equal to $l$ all along the perimeter
-of sensor $j$, we have to ensure that at least $l$ sensors involved in each
-coverage interval $i \in I_j$ of sensor $j$ are active. According to the
-previous notations, the number of active sensors in the coverage interval $i$ of
-sensor $j$ is given by $\sum_{k \in A} a^j_{ik} X_k$. To extend the network
-lifetime, the objective is to activate a minimal number of sensors in each
-period to ensure the desired coverage level. As the number of alive sensors
-decreases, it becomes impossible to reach the desired level of coverage for all
-coverage intervals. Therefore, we use variables $M^j_i$ and $V^j_i$ as a measure
-of the deviation between the desired number of active sensors in a coverage
-interval and the effective number. And we try to minimize these deviations,
-first to force the activation of a minimal number of sensors to ensure the
-desired coverage level, and if the desired level cannot be completely satisfied,
-to reach a coverage level as close as possible to the desired one.
-
-Our coverage optimization problem can then be mathematically expressed as follows:
-%Objective:
-\begin{equation} %\label{eq:ip2r}
-\left \{
-\begin{array}{ll}
-\min \sum_{j \in J} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i )&\\
-\textrm{subject to :}&\\
-\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i \geq l \quad \forall i \in I_j, \forall j \in J\\
-%\label{c1}
-\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i \leq l \quad \forall i \in I_j, \forall j \in J\\
-% \label{c2}
-% \Theta_{p}\in \mathbb{N}, &\forall p \in P\\
-% U_{p} \in \{0,1\}, &\forall p \in P\\
-X_{k} \in \{0,1\}, \forall k \in A
-\end{array}
-\right.
-\notag
+Second, several variables are defined. Hence, each binary variable $X_{k}$ determines the activation of sensor $k$ in the sensing phase ($X_k=1$ if the sensor $k$ is active or 0 otherwise). $M^j_i$ is a variable which measures the undercoverage for the coverage interval $i$ corresponding to sensor~$j$. In the same way, the overcoverage for the same coverage interval is given by the variable $V^j_i$.
+
+To sustain a level of coverage equal to $l$ all along the perimeter of sensor $j$, at least $l$ sensors involved in each coverage interval $i \in I_j$ of sensor $j$ have to be active. According to the previous notations, the number of active sensors in the coverage interval $i$ of sensor $j$ is given by $\sum_{k \in A} a^j_{ik} X_k$. To extend the network lifetime, the objective is to activate a minimal number of sensors in each period to ensure the desired coverage level. As the number of alive sensors decreases, it becomes impossible to reach the desired level of coverage for all coverage intervals. Therefore
+variables $M^j_i$ and $V^j_i$ are introduced as a measure of the deviation between the desired number of active sensors in a coverage interval and the effective number. And we try to minimize these deviations, first to force the activation of a minimal number of sensors to ensure the desired coverage level, and if the desired level cannot be completely satisfied, to reach a coverage level as close as possible to the desired one.
+
+The coverage optimization problem can then be mathematically expressed as follows:
+\begin{equation}
+ \begin{aligned}
+ \text{Minimize } & \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i ) \\
+ \text{Subject to:} & \\
+ & \sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i \geq l \quad \forall i \in I_j, \forall j \in S \\
+ & \sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i \leq l \quad \forall i \in I_j, \forall j \in S \\
+ & X_{k} \in \{0,1\}, \forall k \in A \\
+ & M^j_i, V^j_i \in \mathbb{R}^{+}
+ \end{aligned}
