# Spectrum Decision-Making in Collaborative Cognitive Radio Networks
## 2 Materials and Methods
### Decision-Making Models
#### 2.3.1 Feedback Fuzzy Analytical Hierarchical Process (FFAHP)  
The FFAHP algorithm (Feedback FAHP) proposes the feedback of the information from previous assessments based on the FAHP method [24]. The FAHP algorithm uses fuzzy logic as a tool that is particularly adequate to make decisions in scenarios where the inputs are often uncertain or inaccurate. In essence, the FAHP algorithm uses the same methodology of the AHP algorithm [28]. However, fuzzy logic supports in handling the subjectivity and uncertainty generated during the assessment process [24]. After normalization, the FAHP algorithm states that the weight vector is given by Equation (1), where \(dn\) represents the weights assigned to each criterion. In methods based on the AHP algorithm, judgment matrices must be built that perform comparative assessments that define the level of relative importance between each possible combination of the criteria, sub-criteria, and alternatives--independently [12; 36]. The quality of service is one of the most relevant criteria in decision-making algorithms. It allows differentiating between applications that are delay sensitive and those that are not. In other words, it allows classifying applications with a focus on real time (RT) and best effort (BE). In this investigation, the objective is considered to be the same for RT and BE (selecting the best spectral opportunity). However, the importance (weighing) of each sub-criterion is different, and it corresponds to the focus of the RT and BE applications. For instance, the RT and BE applications are criteria based on different approaches. In RT, the sub-criteria with the highest  
Figure 11: Input and output variable of the multi-criteria decision-making (MCDM) module. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Average** & **Description** \\ \hline AP & Availability probability & Average of each column of the availability matrix \\ AAT & Average availability time & Average of some consecutives of the availability matrix \\ PSINR & Average SINR & Average of each column of the SINR matrix without including zeros \\ ABW & Average bandwidth & Average of each column of the bandwidth matrix \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Vectors for multi-criteria analysis, adapted from [15]. priorities are those that reduce the delay. For BE, the sub-criteria with the highest priority are those that increase the data rate, such as bandwidth and SINR. \[W=\frac{d_{1}}{\sum_{i=1}^{n}d_{i}},\frac{d_{2}}{\sum_{i=1}^{n}d_{i}},\ldots, \frac{d_{n}}{\sum_{i=1}^{n}d_{i}} \tag{1}\]  
If the algorithm is using the RT application, the score for each channel is computed using Equation (2), while Equation (3) is used for BE applications. The coefficients of each of the equations are taken from [24]. Table 9 presents the description of each variable indicated in the equations. \[Score_{RT}=0.3593(AP)+0.2966(AAT)+0.1970(PSINR)+0.1471(ABW) \tag{2}\]  
\[Score_{BE}=0.1607(AP)+0.1523(AAT)+0.3949(PSINR)+0.2921(ABW) \tag{3}\]  
Algorithm 1 presents the programming structure for the FFAHP multi-criteria technique. The algorithm is divided into four blocks: two corresponding to the input and output information, the third block corresponding to the FFAHP multi-criteria technique, and the fourth block corresponding to the ranking. The input block takes the information from the multi-criteria vectors (Table 9) and multiplies it by the coefficients of Equations (2) and (3); this operation allows obtaining a matrix of 4 rows \(\times\) 500 columns. This matrix is taken by the FFAHP block in order to establish the channels (rows) with higher and lower scores. The FFAHP block is conditioned to two scenarios: score assignment for a scenario without feedback and score assignment for a scenario with feedback. For both cases, the ranking block takes the information and organizes the channels in a descending order where the channels with greater spectral opportunity are located in the first positions and those with less spectral opportunity are located in the last positions.