# A Matheuristic for Joint Optimal Power and Scheduling Assignment in DVB-T2 Networks +
## 4 A Matheuristic for DVB Design
### Evolution of the Population
#### 4.2.2 Crossover, Mutation and Death  
The individuals selected in the previous step are randomly associated in couples, so as to form \(\lfloor k\cdot m/2\rfloor\) pairs. After pairing, each couple generates two offspring through chromosome crossover. More in detail, let us consider the crossover of a pair (the _parents_) corresponding with power vectors \(p^{1},p^{2}\); the crossover combines the power levels in the same position of \(p^{1}\), \(p^{2}\) for generating two offspring characterized by new power vectors \(p^{3},p^{4}\) that hopefully are associated with higher fitness value.  
With the aim of evaluating the effect of crossover, we define the measure \(\Delta\mathrm{COV}(p,p_{s}=P_{l})\in\mathbb{Z}\) to represent the variation in the number of covered users that is caused by changing the power value \(p_{s}\) in position \(s\) of a power vector \(p\) to a value \(P_{l}\in\mathcal{P}\), while keeping all other power values unchanged. Exploiting measure \(\Delta\mathrm{COV}(p,p_{s}=P_{l})\), we operate the following crossover, which tries to make \(p^{3}\) the best individual in the offspring. When the crossover starts, \(p^{3}\) and \(p^{4}\) have all elements equal to 0. Then, by following an increasing order of the index \(s\) from 1 to \(|S|\) and, for each \(s\), of the index \(d\) from 1 to \(|D|\), each value 0 inherits the power value of one of the two parents in the same position. More in detail, let us assume to focus on the crossover procedure for a position \((\sigma,\delta)\) with \(\sigma\in\{1,\ldots,|S|\}\) and \(\delta\in\{1,\ldots,|D|\}\): For values of \(s\in\{1,\ldots,|S|\}\) such that \(s<\sigma\) and, for fixed \(\sigma\), for values of \(d\in\{1,\ldots,|D|\}\) such that \(d<\delta\), the crossover has been operated and thus the vectors \(p^{3},p^{4}\) include power levels inherited by the parents \(p^{1},p^{2}\); in contrast, positions corresponding to values \(s\geq\sigma\) and, for fixed \(\sigma\), to values \(d\geq\delta\) have not yet been processed and are thus still equal to 0.  
The inheritance of power values of \(p^{3}\) and \(p^{4}\) from their parents \(p^{1}\) and \(p^{2}\) is set according to these rules:  
\[p^{3}_{sd} = \left\{\begin{array}{ll}p^{1}_{sd}&\mbox{if $\Delta\mathrm{COV}(p^{3},p^{3}_{sd}=p^{1}_{sd})\geq\Delta\mathrm{COV}(p^{3},p ^{3}_{sd}=p^{2}_{sd})$}\\ p^{2}_{sd}&\mbox{otherwise}\end{array}\right.\] \[p^{4}_{sd} = \left\{\begin{array}{ll}p^{1}_{sd}&\mbox{if $\Delta\mathrm{COV}(p^{3},p^{3}_{sd}=p^{1}_{sd})<\Delta\mathrm{COV}(p^{3},p ^{3}_{sd}=p^{2}_{sd})$}\\ p^{2}_{sd}&\mbox{otherwise}\end{array}\right.\]  
which lead \(p^{3}\) to inherit the power values offering the best variation in coverage \(\Delta\mathrm{COV}\).  
In addition to crossover, we also provide for varying the values of power vectors through _mutation_, since it allows the ability to more effectively explore the set of solutions and helps to avoid the trap of local optimal solutions. To implement mutation, at every iteration, we select \(\lfloor\gamma\cdot|POP|\rfloor\) with \(0<\gamma<1\) individuals in a random way. After this, we randomly select \(|L|\cdot|D|\) power levels and we change them to the immediately lower power value included in \(\mathcal{P}\). The aim of this mutation procedure is to create individuals that grant the same coverage but using lower power emission: this is beneficial since it allows the ability to reduce interference among stations. When some crossover or mutation implies a violation of the constraint imposing a maximum difference between powers in adjacent directions of a station, we reduce the power to the closest value in \(\mathcal{P}\) that allows the ability to satisfy the constraint.  
As the last step, some individuals _die_ and are excluded from \(POP\). In particular, we remove the \(2\cdot\lfloor k\cdot m/2\rfloor\) individuals with lowest fitness function values.