# Fuzzy Multi-Criteria Based Trust Management in Heterogeneous Federated Future Internet Testbeds
## 4 Evaluation
### Fuzzy VIKOR Evaluation  
In order to evaluate the fuzzy reputation system, the ratings of K1-K3 are obtained by linguistic terms of Table 2, which are mapped onto triangular fuzzy numbers (A1). The ratings of K4 are numeric and converted to fuzzy numbers according to Step 4 of the modified fuzzy VIKOR methodology. In the following paragraphs, we demonstrate the computations of each step of the fuzzy VIKOR methodology for the NETMODE testbed. The computations for the NITOS testbed are similar, and they are omitted. According to Step 2, the experimenter assigns the pairwise importance for each KPI with respect to the others, so we obtain the PICM,  
\[PICM= \begin{array}{c}K_{1}&K_{2}&K_{3}&K_{4}\\ K_{1}&1&SW&SW&VW\\ K_{2}&SS&1&SW&FW\\ KS_{3}&SS&1&FW\\ V&FS&FS&1\end{array} \]  
\[= \begin{bmatrix}(1,1,1)&(\frac{2}{3},1,1)&(\frac{2}{5},1,1)&(\frac{2}{5},\frac {1}{2},\frac{2}{3})\\ (1,1,\frac{3}{2})&(1,1,1)&(\frac{2}{5},1,1)&(\frac{1}{2},\frac{2}{3},1)\\ (1,1,\frac{3}{2})&(1,1,\frac{3}{2})&(1,1,1)&(\frac{1}{2},\frac{2}{3},1)\\ (\frac{3}{2},2,\frac{5}{2})&(1,\frac{3}{2},2)&(1,\frac{3}{2},2)&(1,1,1)\end{bmatrix} \]  
The consistency ratio of the defuzzified PICM, \(CR=0.018\), is acceptable. As described in Step 3 of Section 3.1, using the fuzzy extended analysis, we obtain the weight vector for the KPIs,  
\[W_{j}= \begin{bmatrix}0.109&0.199&0.233&0.46\end{bmatrix} ,\ j=1,\ldots,\,4\]  
Then, the experimenter evaluates the fuzzy KPIs by using the linguistic variables in Table 2. For the numeric KPI, assume that the credibility mechanism modifies the user's opinion to 0.90; the SLA value was set at 0.80; and the monitoring data for this KPI is 0.85. The triangular membership function of the modified linguistic value \(\widehat{U}\widehat{U}\widehat{U}\) is \(\mu_{\widetilde{X}}=(7.9,8.9,9.9)\). According to the proposed method, we compute the testbed's reputation score for this experiment. The first row of FEM contains the experimenter's ratings for the KPIs, while the ideal ratings of the virtual user are in the second row,  
\[FEM= \begin{array}{c}E\\ V\end{array} \left[ \begin{array}{cccc}E\\ E\end{array} \right.\begin{bmatrix}E\\ E\end{array}\begin{bmatrix}E\\ E\end{array} \begin{bmatrix}\widehat{U}\widehat{U}\widehat{U}\widehat{U}\\ E\end{bmatrix} \]  
The fuzzy best value and fuzzy worst value are determined, and the separation measures are calculated according to (7)-(10). The evaluation indexes \(Q_{i}\) are calculated by (10); \(Q_{1}=0.179\) and \(Q_{2}=0\). Finally, the reputation score for this experiment is computed by (12),  
\[R_{exp}^{T}=(1-Q_{1})\,100\%=82.1\%\]  
The following example illustrates the performance of the reputation system and the credibility mechanism in the case of a suspicious evaluation. Assuming that \(SD=0.8,\ MD=0.9,\ UO=0.6\), the credibility mechanism modifies the user's opinion to \(\widehat{U}\widehat{U}\widehat{U}=0.8055\) with \(\mu_{\widetilde{U}\widehat{U}}=(7.05,8.05,9.05)\), and the user's credibility decreases from 0.8 to 0.756. The final reputation value is computed as 73.9%. This case shows that the user is possibly malicious, and his/her credibility is reduced, while the testbed's reputation score is not significantly affected. The credibility mechanism is important to alleviate the effect of misleading ratings. In the above example, if there were no credibility mechanism, the reputation value would be 66.7%. Finally, it is remarkable that the reputation system based on the modified fuzzy VIKOR is scalable considering the number of KPIs, because the reputation score is computed by simple fuzzy mathematical formulas. In the second scenario, a dataset of two thousand experiments considers any possible case of the experimenter's behavior to highlight the effect of the credibility mechanism and the resilience of the proposed reputation system against malicious users and their ratings. The initial reputation score of the testbed is set to 0.5. In Figure 3, it is shown which case, CASE1-CASE4 of Algorithm 1, is valid for each experiment of the dataset.