# Health Symptom Checking System for Elderly People Using Fuzzy Analytic Hierarchy Process
## 2 MCDA Fuzzy AHP
### Computational Fuzzy AHP Procedure  
Fuzzy AHP comprises of the following steps:  
1. Based on the proposed decision model, the alternatives, criteria, and sub-criteria should be identified.
2. Interrelationships within the structure, including the criteria, sub-criteria, and alternatives, should be determined to construct the hierarchy structure.
3. Establish the TFNs.
4. Construct a fuzzy pairwise comparison matrix of the components with fuzzy judgments.
5. The alternative with the highest overall score was selected as a final priority decision.  
Fuzzy pairwise comparison of TFNs was depicted in Equation (2) based on the decision maker preferences:  
\[\widetilde{O}=\left[\begin{array}{cccc}(o_{11}^{L},o_{11}^{M},o_{11}^{U})&( o_{12}^{L},o_{12}^{M},o_{12}^{U})&\cdots&(o_{1n}^{L},o_{1n}^{M},o_{1n}^{U})\\ (o_{21}^{L},o_{21}^{M},o_{21}^{U})&(o_{22}^{L},o_{22}^{M},o_{22}^{U})&\cdots&(o_ {2n}^{L},o_{2n}^{M},o_{2n}^{U})\\ \vdots&\vdots&\ddots&\vdots\\ (o_{m1}^{L},o_{m1}^{M},o_{m1}^{U})&(o_{m2}^{L},o_{m2}^{M},o_{m2}^{U})&\cdots&(o_ {mn}^{L},o_{mn}^{M},o_{mn}^{U})\end{array}\right] \tag{2}\]  
where \(\widetilde{O}\) is the precise judgment matrix and \(o_{mn}\) represents the comparison of the \(m\)th row element and \(n\)th column element. If \(\widetilde{O}\) was a pairwise comparison matrix, then the fuzzy reciprocal matrix is given as follows.  
\[\widetilde{O}=\left[\begin{array}{cccc}(1,\,1,\,1)&(o_{12}^{L},o_{12}^{M},o_{ 12}^{U})&\cdots&(o_{1n}^{L},o_{1n}^{M},o_{1n}^{U})\\ (\frac{1}{o_{12}^{U}},\frac{1}{o_{12}^{M}},\frac{1}{o_{12}^{L}})&(1,\,1,\,1)& \cdots&(o_{2n}^{L},o_{2n}^{M},o_{2n}^{U})\\ \vdots&\vdots&\ddots&\vdots\\ (\frac{1}{o_{1n}^{U}},\frac{1}{o_{1n}^{M}},\frac{1}{o_{1n}^{L}})&(\frac{1}{o_{ 2n}^{U}},\frac{1}{o_{2n}^{U}},\frac{1}{o_{2n}^{L}})&\cdots&(1,\,1,\,1)\end{array}\right] \tag{3}\]  
Operations of TFN can be described with the following properties [17; 18; 19].  
There are numerous fuzzy operations applied in diverse fields, such as operational research, control theory, and management sciences. We have illustrated those operations applicable in this research below. Suppose that two TFNs O and P are respectively given by O = (o\({}_{1}\), o\({}_{2}\), o\({}_{3}\)) and \(P\) = (p\({}_{1}\), p\({}_{2}\), p\({}_{3}\)). Then, the triangular fuzzy operations are defined as follows:  
* Addition \[[\texttt{o}_{1},\,\texttt{o}_{2},\texttt{o}_{3}]\,+\,[\texttt{p}_{1},\, \texttt{p}_{2},\texttt{p}_{3}]\,=\,[\texttt{o}_{1}+\texttt{p}_{1},\,\texttt{o}_{ 2}+\texttt{p}_{2},\texttt{o}_{3}+\texttt{p}_{3}]\] (4)
* Multiplication \[[\texttt{o}_{1},\,\texttt{o}_{2},\texttt{o}_{3}]\,*\,[\texttt{p}_{1},\, \texttt{p}_{2},\texttt{p}_{3}]\,=\,[\texttt{o}_{1}*\texttt{p}_{1},\,\texttt{o}_{ 2}*\texttt{p}_{2},\texttt{o}_{3}*\texttt{p}_{3}]\] (5)
* Division \[[\texttt{o}_{1},\,\texttt{o}_{2},\texttt{o}_{3}]\,/\,[\texttt{p}_{1},\,\texttt{p} _{2},\texttt{p}_{3}]\,=\,[\frac{\texttt{o}_{1}}{\texttt{p}_{3}},\frac{\texttt{o}_{ 2}}{\texttt{p}_{2}},\frac{\texttt{o}_{3}}{\texttt{p}_{1}}]\] (6)
* Inverse \[[\texttt{o}_{1},\,\texttt{o}_{2},\texttt{o}_{3}]^{-1}\,=\,[\frac{1}{\texttt{o}_{3}}, \frac{1}{\texttt{o}_{2}}\,\,\texttt{,}\,\frac{1}{\texttt{o}_{1}}]\] (7)  
Fuzzy numbers are described in linguistic concepts with words or sentences in natural or artificial language, such as low, medium, and high in a particular context, to express the patient's feeling. The linguistic variable is used to deal with the complexity of creating the importance of preference scale for criteria and alternatives instead of the crisp values in fuzzy AHP [20; 21; 22].