# Health Symptom Checking System for Elderly People Using Fuzzy Analytic Hierarchy Process
## 3 Approach
### Weight Determination for Various Diseases
If \(b_{ij}\) was the element of row \(i\) and column \(j\) of the matrix, then the lower diagonal elements were filled as follows:  
\[b_{ji}=\frac{1}{b_{ij}};\;i,\;j\;=\;1,\;2,\;\ldots,\;n. \tag{8}\]  
Thus, the lower triangular matrix comparison can be completed as shown in Table 3.  
Figure 5: The structural depiction of symptom checker AHP. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & **Chest Pain** & **Toothache** & **Fever** & **Thist** & **Weight Loss** & **Blurred Vision** & **Dizziness** & **Swelling** & **Depression** \\ \hline
**Chest Pain** & \(1,1,1\) & 0.02, 0.18, 0.80 & 0.07, 0.23, 0.70 & 0.13, 0.27, 0.60 & 0.13, 0.27, 0.60 & 0.13, 0.27, 0.60 & 0.07, 0.23, 0.70 & 0.33, 0.27, 0.40 & 0.22, 0.28, 0.50 \\
**Toothache** & & \(1,1,1\) & 0.02, 0.28, 0.50 & 0.07, 0.23, 0.70 & 0.02, 0.18, 0.80 & 0.07, 0.23, 0.70 & 0.13, 0.27, 0.60 & 0.80, 0.10, 0.10 & 0.07, 0.23, 0.70 \\
**Fever** & & & & & & & & & \\
**Thirst** & & & & & & & & & \\
**Wight Loss** & & & & & & & & & & \\
**Blurred Vision** & & & & & & & & & & \\
**Dizziness** & & & & & & & & & & \\
**Swelling** & & & & & & & & & & \\
**Depression** & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pairwise comparison matrix of upper triangular criteria symptoms. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & **Chest Pain** & **Toothache** & **Fever** & **Thist** & **Weight Loss** & **Blurred Vision** & **Dizziness** & **Swelling** & **Depression** \\ \hline
**Chest Pain** & \(1,1,1\) & 0.02, 0.18, 0.80 & 0.07, 0.23, 0.70 & 0.13, 0.27, 0.60 & 0.13, 0.27, 0.60 & 0.13, 0.27, 0.60 & 0.07, 0.23, 0.70 & 0.33, 0.27, 0.40 & 0.22, 0.28, 0.50 \\
**Toothache** & & \(1,1,1\) & 0.02, 0.28, 0.50 & 0.07, 0.23, 0.70 & 0.02, 0.18, 0.80 & 0.07, 0.23, 0.70 & 0.13, 0.27, 0.60 & 0.80, 0.10, 0.10 & 0.07, 0.23, 0.70 \\
**Fever** & & & & & & & & & \\
**Thirst** & & & & & & & & & \\
**Wight Loss** & & & & & & & & & \\
**Blurred Vision** & & & & & & & & & \\
**Dizziness** & & & & & & & & & \\
**Swelling** & & & & & & & & & \\
**Depression** & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Complete pairwise comparison matrix of criteria symptoms. Then a normalized matrix was created after full comparisons of the matrix as follows:  
\[k_{ij}=\frac{b_{ij}}{\sum\limits_{p=1}^{m}b_{pj}} \tag{9}\]  
To evaluate the weights for each criterion and for each alternative with reference to a given criterion, the fuzzy synthetic extent analysis method was adopted here [31]. \(A_{i}\) with respect to \(i\)th criteria was achieved, where \(C\) was the set of criteria and \(n\) was the number of criteria, \(m\) was the number of decision alternatives and the subscripts \(i\) and \(j\) represented rows and columns, respectively. \[\text{A}_{i}=\sum\limits_{j=1}^{m}M_{C_{i}}^{j}\left[\sum\limits_{i=1}^{n} \cdot\prod\limits_{j=1}^{m}M_{C_{i}}^{j}\right]^{-1} \tag{10}\]  
After obtaining the weights of the different criteria of symptoms associated with the fuzzy synthetic extent of TFNs, the relative weights were calculated as follows:  
\[w_{i}=\frac{w_{i}}{\sum_{j=1}^{m}w_{j}},i=1,2,...,m. \tag{11}\]  
where \(w_{i}\) represented the normalized weight for the \(i\)th criterion. Table 4 showed the synthetic extent values and normalized weights for different criteria. A similar mathematical methodology for weight calculation was used to estimate the weights for the given alternatives _diseases_ with respect to criteria _symptoms_, and four of their respective results are shown in Tables 5-8. The possibility of final priority weight for different diseases with respect to the symptoms were calculated as follows:  
\[Weight=\sum p_{ij}*w_{i};\;i=1,\;2,\ldots,m,\;j=1,\;2,\ldots,n. \tag{12}\]  
where \(p_{ij}\) was the weight for diseases with respect to symptoms by pairwise comparison from Tables 5-8. In Equation (12), \(w_{i}\) represents the relative weights for different criteria symptoms by pairwise comparison from Table 4. Table 9 shows the priority weights for different diseases.