# Occupational Risk Assessment of Wind Turbines in Bangladesh
## 4 Research Methodology
### Fuzzy TOPSIS Method  
Fuzzy TOPSIS multi-criteria is a higher cognitive process and has been credited with helping to create many methods to solve multi-criteria problems [31]. The principle of TOPSIS is that the choice must be the closest to the most preferred solution and the furthest from the least preferred outcome. The most preferred outcome should maximize the benefits and reduce the costs, while the least preferred outcome should have the opposite effect. In order to achieve maximum benefit, the option closest to the most preferred outcome should be chosen. FTOPSIS methods were developed after the fuzzy set theory [32]. The FTOPSIS method is as follows [7;33]:  
Step 1: Determine the weighted value of each criterion. This analysis employs fuzzy AHP to search out the fuzzy preference weights. Step 2: Construct the fuzzy performance/decision matrix and choose the suitable linguistic variables for the alternatives concerning criteria:  
\[\widetilde{D}= \begin{array}{ccccc}&C_{1}&C_{2}&\ldots&&C_{n}\\ &A_{1}&\left[\begin{array}{ccccc}\widetilde{x}_{11}&\widetilde{x}_{12}& \cdots&\widetilde{x}_{1n}\\ \widetilde{x}_{21}&\widetilde{x}_{22}&\cdots&\widetilde{x}_{2n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \widetilde{x}_{m1}&\widetilde{x}_{m2}&\cdots&\widetilde{x}_{mn}\end{array} \right]\\ \end{array}  \tag{5}\]  
where \(i=1,\,2,\,3,\,\ldots\,\ldots\,\,\ldots\,\,m\); \(j=1,\,2,\,3,\,4,\,\ldots\,\,\ldots\,\,\ldots\,\,n\);  
\[\text{and}\ \widetilde{X}_{ij}=\frac{1}{K}\ (\widetilde{x}_{ij}^{k}\ \oplus\ \ldots\ldots\oplus\widetilde{x}_{ij}^{l}\ldots\ldots\oplus\widetilde{x}_{ij} ^{k})\]  
where \(\widetilde{x}_{ij}^{k}\) is the performance of alternative \(A_{i}\) concerning criterion \(C_{j}\) according to the data of \(k^{th}\) expert, and \(\widetilde{X}_{ij}^{k}=\left( \begin{array}{ccccc}k_{ij}^{k},&m_{ij}^{k},&u_{ ij}^{k}\end{array} \right)\). Step 3: Develop a combined decision matrix using the following formula:  
\[\widetilde{x}_{ij}\!\!=\left(l_{ij},&m_{ij},&u_{ij}\right)\ \text{where}\ l_{ij}=\min \!\left\{l_{ij}^{k}\right\},\ m_{ij}=\frac{1}{k}\ \sum_{k=1}^{k}m_{ij}^{k},\ \ u_{ij}=\max\!\left\{u_{ij}^{k}\right\} \tag{6}\]  
Step 4: Normalize the fuzzy decision matrix by the following formula:  
\[\widetilde{R}=\left[\widetilde{r}_{ij}\right]_{m\times n^{\prime}}\text{, where}\ i=1,\,2,\,3,\,\ldots\ldots\,,\ m;\ j=1,\,2,\,3,\,\ldots,\,n \tag{7}\]  
and \(\widetilde{r}_{ij}=\left( \begin{array}{ccccc}l_{ij}&m_{ij}\\ \hline u_{ij}^{k}&\frac{m_{ij}}{u_{j}^{k}}&\frac{u_{ij}}{u_{j}^{k}}\end{array} \right)\), where \(u_{j}^{+}=max_{i}\!\left\{u_{ij}|i=1,\,2,\,3,\,\ldots\ldots\ldots,\,\,n\,\,\right\}\). The weighted fuzzy normalized decision matrix is shown as the following matrix \(\widetilde{V}\)  
\[\text{where the }\widetilde{V}\!\!=\left[\widetilde{v}_{ij}\right]_{n\times m^{ \prime}},\ \ \ i=1,\,2,\,3,\,\ldots,m;j=1,\,2,\,3,\ldots,\,n \tag{8}\]and \(\widetilde{v}_{ij}=\widetilde{r}_{ij}\otimes\ \widetilde{w}_{j}\); here, \(\widetilde{w}i\) is the fuzzy weight of criterion \(i\). Step 5: Evaluate the fuzzy positive ideal solution (FPIS) and fuzzy negative solution (FNIS). From the weight-normalized fuzzy decision matrix, it is clear that the elements \(\widetilde{v}_{ij}\) are normalized positive triangular fuzzy numbers. Its range is [0, 1]. The fuzzy positive ideal solution \(A^{+}\) and fuzzy negative ideal solution \(A^{-}\) are as follows:  
\[A^{+}=\left(\widetilde{v}_{1}^{*},\ldots\qquad\widetilde{v}_{j}^{*},\ldots.\ \ \ \widehat{v}_{n}^{*}\right) \tag{9}\]  
\[A^{-}=\left(\widetilde{v}_{1}^{-},\ldots\qquad\widetilde{v}_{j}^{-},\ldots.\ \ \ \widehat{v}_{n}^{-}\right) \tag{10}\]  
where \(\widetilde{v}_{1}^{*}=(1,1,1)\otimes\widetilde{w}_{j}=(lv_{j},\ mw_{j},\ uw_{j})\) and \(\widetilde{v}_{1}^{-}=(0,\ 0,\ 0)\), \(j=1,\ 2,\ 3,\ \ldots\ldots,n\). Step 6: Calculate the difference \((d_{i}^{+},\ d_{i}^{-})\) between each criterion and fuzzy positive ideal solution as well as a negative ideal solution.