# A Matheuristic for Joint Optimal Power and Scheduling Assignment in DVB-T2 Networks +
## 2 DVB-T2 Network Design  
In this section, we derive the mixed integer linear programming model that we use as basis for our new matheuristic algorithmic proposal. To this end, we first describe the essential elements constituting the DVB-T2 network that we consider: the network is made up of a set of DVB-T transmitting stations \(S\) that have the task of broadcasting digital television signals to users located in a region. Each station \(s\in S\) is characterized by a number of parameters: primarily, its location and then its radio-electrical parameters, such as the emitted power and the channel used to transmit. In accordance with recommendations and regulations released by telecommunication authorities, like the Italian Authority for Telecommunications (AGCOM) [32], the region to be covered with DVB-T services is partitioned into a raster of pixels corresponding to small squared portions of territories. The center of each pixel is called a _testpoint_ (TP) and is taken as representative of all points in the pixel, in particular concerning propagation behavior of signals throughout the pixel. The set of all testpoints included in the region is denoted by \(T\). The Wireless Network Design problem (WND) is an optimization problem that generally consists of choosing the parameters of a set of wireless transmitters (e.g., their location and radio-electrical configuration) with the aim of providing service coverage to a set of users while maximizing a revenue function (see e.g., [13; 14; 26; 33]). Notwithstanding the quite high number of parameters that characterize a transmitter and that can be configured, all works considering the WND focuses just on a subset of them. In particular, the vast majority of works include the setting of the power emission of transmitters and the assignment of served users to deployed transmitters as decision variables, since they constitute critical decision aspects in the design of a wireless network (see, for example, [7; 26; 27; 34] related to DVB, [35; 36] related to 5G, [23; 37] related to FTTx, [38; 39] related to mesh networks, [40; 41] related to UMTS, [33; 42] related to WiMAX, [43; 44] related to WLANs and [13; 14; 45; 46; 47; 48] related to other kind of wireless network technologies). If the optimization problem requires to decide the power emitted by transmitters and to choose the assignment of users to transmitters, it is obtained the so-called Scheduling and Power Assignment Problem (SPAP), which is NP-hard [27] and is considered a crucial WND problem in a hierarchy that has been developed in [27; 33; 49]. Before introducing the decision variables that are included in the optimization model, we make additional considerations about the power emission of DVB-T stations: such stations typically correspond with very powerful transmitters that have the task of covering vast portions of territory and host antennas whose horizontal radiation patterns can be customized (see e.g., [50]). Such customization of the horizontal antenna diagram typically considers 36 directions associated with angles from 0 to 350 degrees separated by a 10-degree interval and, from a modeling point of view, can be represented as the possibility of specifying the power emitted by a station \(s\in S\) in each direction \(d\in D=\{1,2,\ldots,36\}\). As noted in [50], the power emitted by the same station in different directions cannot exhibit too large differences. In particular, the difference between adjacent directions cannot exceed a threshold denote by \(\Delta P\). For each station \(s\in S\) and TP \(t\in T\), we denote by \(\delta(s,t)\in D\) the (unique) direction on which station \(s\) emits signals to \(t\).