# Fuzzy Multi-Criteria Based Trust Management in Heterogeneous Federated Future Internet Testbeds
## 3 Proposed Fuzzy VIKOR Reputation System
### Fuzzy VIKOR  
Fuzzy VIKOR is a multi-criteria decision-making approach that simultaneously measures the closeness to the best and worst alternative. It can be applied to any scenario in which a user has to select among alternative providers, such as cloud services [26] and renewable energy resources [25]. The original fuzzy VIKOR approach uses explicitly a group of fuzzy KPIs. For the computation of the reputation score of a testbed, we extend this approach in order to process also numeric KPIs, as shown in Figure 1, where the reputation value is directly derived from the KPIs. In Figure 1, the level of criteria categories does not contribute to the computation of the reputation value, and it indicates only the different nature of the underlying KPIs. The purple (left) technical KPIs refer to QoS metrics, and they are numeric, while the pink (right) non-technical KPIs correspond to fuzzy KPIs. These numeric inputs are converted to fuzzy numbers, as explained in the following subsection. For each pair of KPIs, an assigned fuzzy weight indicates the relative importance between them. Table 1 presents the linguistic terms and the corresponding membership functions for the fuzzy weights, while Table 2 presents the information about the fuzzy numbers used for the KPIs' evaluation. The experimenter's evaluation is compared to the perfect evaluation of a virtual user in order to acquire a quantitative measure of the closeness to the best testbed's performance. The following steps include all the necessary computations of the reputation score of a federated testbed. Step 1. Definition of the testbed KPIs: The testbed provider defines all the QoS and QoE KPIs that determine the performance of the testbed. Furthermore, the QoS KPIs are included in an SLA between the provider and the experimenter. Step 2. Definition of relative importance weights: The experimenter assigns the linguistic term for the relative importance weight of all possible KPI pairs from Table 1. Assuming a testbed with \(N\) KPIs, we formulate the fuzzy Pairwise Importance Comparison Matrix (PICM) as follows:  
\[ \begin{array}{c}\text{\emph{PICM}}=\begin{array}{c}\text{\emph{K}}_{1}\\ \text{\emph{K}}_{2}\\ \text{\emph{K}}_{N}\\ \text{\emph{K}}_{N}\end{array}\begin{bmatrix}K_{1}&K_{2}&\ldots&\text{\emph{K}} _{N}\\ \text{\emph{K}}_{1}&FS&\cdots&\text{\emph{V}}W\\ \text{\emph{FW}}&1&\cdots&\text{\emph{E}}\\ \text{\emph{FW}}&\vdots&\vdots&\ddots&\vdots\\ \text{\emph{V}}S&\text{\emph{E}}&\cdots&1\end{bmatrix}\end{array}  \tag{1}\]  
Then, the PICM's elements are defuzzified using (A6) of Appendix A. Since the weights are derived from the subjective preferences of individuals, the final computation of reputation can be based on inconsistent and conflicting KPIs. Thus, in order to avoid such inconsistencies, the Consistency Ratio (CR) [27] is calculated for each group of sibling attributes. The CR is the degree of randomness in the weight assignment between several sibling attributes. CR values less than 0.1 are acceptable to continue to the next phase; otherwise, the experimenter must correct the assigned weights. Step 3. Computation of the KPIs weight vector: The fuzzy Analytical Hierarchical Process (AHP) is a methodology for determining the relative importance of the selection criteria. In our approach, the extended analysis on fuzzy AHP, as proposed by [PERSON] [28], is adopted to determine the KPIs' weight vector. The following steps of extent analysis on fuzzy AHP are applied. Let the N-dimensional fuzzy \(\text{\emph{PICM}}= \begin{bmatrix}a_{ij}\end{bmatrix} \), \(i,j=1,\ldots,N\); the fuzzy synthetic extent is defined by,  
\[D_{i}= \begin{pmatrix}D_{i}^{l},D_{i}^{m},D_{i}^{u}\end{pmatrix} =\sum_{j=1}^{N}a _{ij}\otimes\left(\sum_{i=1}^{N}\sum_{j=1}^{N}a_{ij}\right)^{-1} \tag{2}\]  
We find the attribute with the higher fuzzy synthetic degree by computing the degree of possibility for a fuzzy number to be greater than other one,  
\[V \begin{pmatrix}D_{i}\geq D_{j}\end{pmatrix} =bgt \begin{pmatrix}D_{i}\cap D_{j} \end{pmatrix} \]  
\[=