# Modeling Trust in IoT Systems for Drinking-Water Management
## 3 Methods and Proposed Approach
### Proposed Approach
#### 3.2.2 Formalization of Trust Calculation  
\[C_{ij}=\sum_{k=1}^{m}W_{ijk}\times F_{ijk} \tag{1}\]  
where  
* \(m\) is the number of factors considered. * \(F_{ijk}\) is the value of factor \(k\) for the relationship between \(i\) and \(j\). This factor is a specific characteristic influencing trust between these critical elements, chosen according to ISO 8000-61 standard to ensure an objective selection [29]. * \(W_{ijk}\) is the weight associated with factor \(k\) for the relationship between \(i\) and \(j\), calculated using the (AHP) method. AHP is a method that decomposes a complex problem into a hierarchy of criteria and sub-criteria and then evaluates and compares these \begin{table}
\begin{tabular}{c c c} \hline \hline
**Incident** & **Description** & **Response** \\ \hline Water Contamination & Detection of contaminants in treated water beyond & Immediate cessation of distribution of contaminated water. \\ Detected & acceptable thresholds. & Investigation, corrective measures, and reinforcement of quality control \\ Cybersecurity Attack & Attempted attack targeting & Activation of enhanced security protocols. Isolation of affected systems and preventive measures to avoid future attacks. \\ Ozonizer Malfunction & Issues in ozone production compromising the efficiency of the ozonation process. & Temporary suspension of ozonation. \\  & & Swift repair or replacement of defective equipment. Strengthening of preventive maintenance procedures. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._elements quantitatively. We will describe it clearly in the next section. The goal is to assign values to different factors to enhance the objectivity of our approach [30]:  
1. **Calculation of Weights:** Let \(n\) be the number of criteria. We will compare each criterion with every other in terms of their relative importance. Comparisons are made on a scale, typically from 1 to 9, where 1 means \"equivalent\" and 9 means \"extremely more important\". This matrix is symmetric. \[A= \begin{bmatrix}1&a_{12}&\dots&a_{1n}\\ \frac{1}{a_{12}}&1&\dots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{1}{a_{1n}}&\frac{1}{a_{2n}}&\dots&1\end{bmatrix} \] (2) where \(a_{ij}\) is the measure of the relative importance between criteria \(i\) and \(j\). 2. **Calculation of Relative Weights:** Calculate the relative weights for each criterion by averaging the columns of the comparison matrix. \[W_{i}=\frac{1}{n}\sum_{j=1}^{n}a_{ij}\] (3) where \(W_{i}\) is the relative weight for factor \(i\). 3. **Normalization of Relative Weights:** Normalize the relative weights to obtain normalized weights. This ensures that the sum of all weights is equal to 1 (normalizing weights to 1 in AHP is a common practice to make results more consistent, interpretable, and comparable, facilitating the decision-making process). \[w_{i}=\frac{W_{i}}{\sum_{i=1}^{n}W_{i}}\] (4) where \(w_{i}\) is the normalized weight for criterion \(i\). 4. **Calculation of Aggregated Weights:** Aggregated weights are calculated by multiplying the normalized weights of the lower levels by the relative weights of the higher levels and summing these products for each criterion. \[W_{\text{aggregated},i}=\sum_{j=1}^{n}W_{j}\cdot w_{ij}\] (5) where \(W_{\text{aggregated},i}\) is the aggregated weight for criterion \(i\). This model provides flexibility in considering various factors influencing trust, and the weights allow for the adjustment of the impact of each factor. It is a general and objective representation that is adaptable to specific conditions of water-treatment stations, such as Sherbrooke's, by integrating the unique characteristics of each critical element and relationship. The following subsection presents a formalization of trust calculation within the context of water-treatment plant management. This formalization relies on specifically chosen functions to address the unique needs of this crucial domain. Let us explore why these functions were selected and how they align with the dedicated purpose of water-treatment management.