# Fuzzy Multi-Criteria Based Trust Management in Heterogeneous Federated Future Internet Testbeds
## 3 Proposed Fuzzy VIKOR Reputation System
### Fuzzy VIKOR
\begin{cases}1&\text{if }D_{i}^{m}\geq D_{j}^{m}\\ \frac{D_{i}^{l}-D_{i}^{u}}{\left(D_{i}^{m}-D_{i}^{u}\right)-\left(D_{j}^{m}-D_{ j}^{l}\right)}&\text{if }D_{i}^{m}\leq D_{j}^{m}\text{\emph{ and }}D_{j}^{l}\leq D_{i}^{u}\\ 0&\text{otherwise}\end{cases}  \tag{3}\]  
The degree of possibility that a fuzzy synthetic extent \(D_{i}\) is greater than the remaining synthetic fuzzy extents of the fuzzy PICM is,  
\[d_{i}=V\left(D_{i}\geq D_{k},\forall k=1,\ldots,N,k\
eq i\right)=\min V \begin{pmatrix}D_{i}\geq D_{j}\end{pmatrix}  \tag{4}\]  
Finally, the normalized weight vector of KPIs is obtained,  
\[W= \begin{bmatrix}w_{1}\ldots w_{N}\end{bmatrix} \text{where }w_{i}=\frac{d_{i}}{\sum_{k=1}^{N}d_{k}} \tag{5}\]  
Step 4. Evaluation of an experiment: Upon the completion of an experiment, the user is prompted to submit his/her judgment of the performance of the testbeds used. In order to mitigate the effect of malicious ratings, the experimenter's credibility is considered. Thus, for the QoS KPIs, the user's opinion \(x\) is properly modified to \(\tilde{x}\) by the credibility mechanism of the following subsection. For the fuzzy KPIs, the experimenter evaluates the testbeds using the linguistic values of Table 2. For the numeric KPIs, the user assigns crisp values, which are converted to fuzzy numbers usingthe membership functions of Table 2. Assume that the numeric evaluation \(\tilde{x}\), modified by the credibility mechanism, corresponds to two adjacent linguistic values \(A\) and \(B\) and \(\mu_{A}\) and \(\mu_{B}\) are the respective membership functions. Then, the modified linguistic value \(\tilde{X}\) that corresponds to the numeric evaluation is obtained by \(\tilde{X}=\mu_{A}A+\mu_{B}B\). We use a virtual user's rating with Excellent linguistic values (E) for all KPIs in order to represent the best possible performance of the testbed. Thus, the obtained Fuzzy Evaluation Matrix (FEM) for the conducted experiment is,  
\[FEM= \begin{array}{cccc}&K_{1}&K_{2}&\cdots&K_{N}\\ &U&\left[\begin{matrix}\tilde{x_{1}}&\tilde{x_{2}}&\cdots&\tilde{x_{N}}\\ E&E&\cdots&E\end{matrix}\right]\end{array}  \tag{6}\]  
where the first row of FEM corresponds to the modified experimenter's (U) opinion, while the second row corresponds to the perfect ratings of the Virtual user (V). Step 5. Computation and update of reputation: In this step, the modified fuzzy VIKOR method is actually applied. Let the weight vector \(W\) and the fuzzy evaluation matrix \(FEM=[x_{ij}],\ i=1,2\), \(j=1,\ldots,N\); we determine the fuzzy best value \(\tilde{f}_{j}^{+}\) and the fuzzy worst value \(\tilde{f}_{j}^{-}\). Since the virtual user ratings are perfect, the fuzzy best and worst values are,  
\[\tilde{f}_{j}^{+}=x_{2j},\ j=1,\ldots,N\] \[\tilde{f}_{j}^{-}=x_{1j},\ j=1,\ldots,N\]  
Then, the separation measure of \(x_{ij}\) from the fuzzy best and worst value is obtained by,  
\[\tilde{S}_{i}=\frac{\sum\limits_{j=1}^{N}w_{j}\left(\tilde{f}_{j}^{+}-x_{ij} \right)}{\tilde{f}_{j,u}^{+}-\tilde{f}_{j,l}^{-}} \tag{7}\]  
\[\tilde{R}_{i}=\max_{j}\left[\frac{w_{j}\left(\tilde{f}_{j}^{+}-x_{ij}\right)} {\tilde{f}_{j,u}^{+}-\tilde{f}_{j,l}^{-}}\right] \tag{8}\]  
Next, the best and worst values of \(\tilde{S}_{i},\ \tilde{R}_{i}\) are calculated,  
\[\tilde{S}^{+}=\min_{i}\tilde{S}_{i},\ \tilde{S}^{-}=\max_{i}\tilde{S}_{i} \tag{9}\]  
\[\tilde{R}^{+}=\min_{i}\tilde{R}_{i},\ \tilde{R}^{-}=\max_{i}\tilde{R}_{i} \tag{10}\]  
Then, the evaluation index \(\tilde{Q}_{i}\) contains the fuzzy reputation score of the experimenter and the virtual user and is computed as,  
\[\tilde{Q}_{i}=\alpha\frac{\tilde{S}_{i}-\tilde{S}^{+}}{\tilde{S}^{-}-\tilde{S }^{+}}+\left(1-\alpha\right)\frac{\tilde{R}_{i}-\tilde{R}^{+}}{\tilde{R}^{-}- \tilde{R}^{+}} \tag{11}\]  
where \(\alpha\) is an index of our willingness to penalize the poor testbed performance or reward the good one; in order to have a balance between good and poor behavior. We set \(\alpha=0.4\), because the virtual user always has excellent ratings.