## 4 Case Studies
### Another Example of Network Security System Assessment
#### 4.2.1 Use the Assessment Approach Proposed in This Paper to Assess Network Security Systems  
There are four alternative network security systems from different information technology companies, denoted as \(A_{i}\) (\(i=1\), \(2\), \(3\), \(4\)), for the military to select. The purpose of assessing these network security systems is to assist the decision-maker in making the best choice. The attributes used to evaluate these computer network security systems are denoted as \(G_{i}\) (\(i=1\), \(2\), \(3\), \(4\), \(5\)). Figure 4: Comparison of evaluation results before and after changing \(m_{15}\). Figure 5: Comparison of evaluation results before and after changing evidence \(m_{22}\). They are tactics, technology, economy, logistics and strategy, respectively, and their weight vector is \(\omega=\left\{0.1,\,0.15,\,0.2,\,0.3,\,0.25\right\}\). There are three decision-makers, denoted as \(R_{i}\left(i=1,\,2,\,3\right)\), and their weight vector is \(\lambda=\left\{0.3294,\,0.3365,\,0.3341\right\}\). The linguistic term set \(S\) is defined as \(S=\left\{s_{0}=extremely\,\,poor(EP),\,s_{1}=very\,\,poor(VP),\,s_{2}= poor(P),\,s_{3}=medium(M),\,s_{4}=good(G),\,s_{5}=very\,\,good(VG),\,s_{6}=extremely\,\,good(EG)\right\}\). The four possible alternatives \(A_{i}\left(i=1,2,3,4\right)\) are to be evaluated using the linguistic term set \(S\) by the three decision-makers under the above five attributes, and construct the decision matrices \(R_{k}=\left(r_{ij}^{\left(k\right)}\right)_{4\times 5}\left(k=1,\,2,\,3\right)\) as follows: \[R_{1}= \begin{array}{c}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{array} \left[ \begin{array}{cccc}S_{5}&S_{6}&S_{3}&S_{4}&S_{6}\\ S_{3}&S_{1}&S_{2}&S_{1}&S_{0}\\ S_{4}&S_{0}&S_{6}&S_{3}&S_{2}\\ S_{1}&S_{5}&S_{3}&S_{2}&S_{3}\end{array} \right]\] \[R_{2}= \begin{array}{c}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{array} \left[ \begin{array}{cccc}S_{3}&S_{4}&S_{2}&S_{6}&S_{2}\\ S_{6}&S_{6}&S_{5}&S_{3}&S_{1}\\ S_{2}&S_{3}&S_{6}&S_{1}&S_{3}\\ S_{1}&S_{0}&S_{4}&S_{4}&S_{6}\end{array} \right]\] The method to convert the decision matrices into BPAs is as follows. * According to the weights of three decision-makers, the evaluation data based on linguistic information are transformed into the probability distribution of linguistic variables. * By applying the uncertainty measure \(iTU^{I}\), the uncertainty of the probability distribution obtained in the previous step can be derived. Then, the uncertainty is used to discount the probability distribution to generate BPAs for evaluation. The following gives an example to clearly illustrate the process of generating BPA for evaluation. According to the decision matrices, for \(A_{1}\), the assessment of its desirability level under \(G_{1}\) given by three decision-makers \(R_{i}\left(i=1,\,2,\,3\right)\) is \(s_{5},s_{3}\) and \(s_{1}\), respectively. Then, the probability distribution of \(A_{1}\) under \(G_{1}\) is defined as:  
\[ \begin{array}{c}m\left(s_{1}\right)=0.3341\\ m\left(s_{3}\right)=0.3365\\ m\left(s_{5}\right)=0.3294\end{array} \]  
By using Equation (13), the uncertainty of the probability distribution can be calculated as \(0.1091\). Let \(\alpha=1-0.1091=0.8909\). Then, the final BPA for evaluation is defined as:  
\[ \begin{array}{c}m\left(s_{1}\right)=\alpha\times m\left(s_{1}\right)=0.2976 \\ m\left(s_{3}\right)=\alpha\times m\left(s_{3}\right)=0.2998\\ m\left(s_{5}\right)=\alpha\times m\left(s_{5}\right)=0.2935\\ m\left(\left\{s_{1},s_{3},s_{5}\right\}\right)=\alpha=0.1091\end{array} \]  
In order to rank these four network security systems, defuzzification is performed to get the total score for each network security system in this example.