# A Matheuristic for Joint Optimal Power and Scheduling Assignment in DVB-T2 Networks +
## 2 DVB-T2 Network Design
By means of simple linear algebra operations, we can reorganize Equation (1) into the inequality that follows, which is usually named _SIR inequality_:  
\[\sum_{\sigma\in U(s,t)}a_{t\sigma}\cdot p_{\sigma\;\delta(\sigma,t)}-\phi\sum_ {\sigma\in I(s,t)}a_{t\sigma}\cdot p_{\sigma\;\delta(\sigma,t)}\ \geq\ \phi\cdot N\;. \tag{2}\]  
As one of the decisions to be considered in the problem is establishing which is the serving station of a TP, we must include one SIR inequality for every potential serving station \(s\in S\) of every TP \(t\in T\). As a consequence, for every \(t\), we face the following disjunctive constraint that contains one SIR inequality for each station \(s\):  
\[\bigvee_{s\in S}\left(\sum_{\sigma\in U(s,t)}a_{t\sigma}\cdot p_{\sigma\; \delta(\sigma,t)}-\phi\sum_{\sigma\in I(s,t)}a_{t\sigma}\cdot p_{\sigma\; \delta(\sigma,t)}\ \geq\ \phi\cdot N\right)\;. \tag{3}\]  
The previous single disjunctive constraint can be reformulated as a set of linear constraints, according to a standard MILP approach (see, for example, [26; 53]), which is based on defining a large constant \(M\) (commonly known as _big-M coefficient_) and exploiting the service variable \(x_{ts}\) as a binary variable that activates or deactivates the constraint. More in detail, Equation (3) can be replaced with the following set of modified SIR inequalities (one for each potential server \(s\in S\)) that also include the big-M coefficient and the variables \(x_{ts}\):  
\[\sum_{\sigma\in U(s,t)}a_{t\sigma}\cdot p_{\sigma\;\delta(\sigma,t)}-\phi\sum_ {\sigma\in I(s,t)}a_{t\sigma}\cdot p_{\sigma\;\delta(\sigma,t)}+M(1-x_{ts})\ \geq\ \phi\cdot N\;. \tag{4}\]  
We call such modified inequality _SIR constraint_ and we remark that it represents a fundamental component of every WND problem that considers coverage of users by means of SIR quantities. It is straightforward to verify that in Equation (4), when \(x_{ts}=1\), we obtain the original SIR inequality without a big-M coefficient, and such an inequality must be fulfilled, thus it is required to serve TP \(t\) through station \(s\) as a server. Instead, when \(x_{ts}=0\), the large value of the big-M coefficient is added to the left-hand-side of the SIR inequality, making it satisfied for any setting of power variables of the stations and thus making the constraint redundant. We provide an overview of the system elements and notation introduced above in Table 1.  
We can then proceed to define the problem of designing a DVB network as follows. \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Notation** \\ \hline \(S\) & set of DVB stations \\ \(T\) & set of testpoints (TPs) \\ \(D\) & set of directions \\ \(\delta(\sigma,t)\) & direction on which station \(s\) emits signals to TP \(t\) \\ \(P_{\max}\) & maximum power emission of a station \\ \(\Delta P\) & maximum power difference between adjacent directions of a station \\ \(a_{ts}\) & fading coefficient of station \(s\) emitting to TP \(t\) \\ \(U(s,t)\) & subset of stations emitting useful signals when station \(s\) serves TP \(t\) \\ \(I(s,t)\) & subset of stations emitting interfering signals when station \(s\) serves TP \(t\) \\ \(\phi\) & SIR threshold \\ \(N\) & system noise \\ \(M\) & big-M coefficient \\ \(p_{sd}\) & continuous decision variable representing the power emission of station \(s\) in direction \(d\) \\ \(x_{ts}\) & binary decision variable representing whether TP \(t\) is served by station \(s\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System elements and notation.