## 2 Materials and Methods
### Material Selection
#### 2.2.2 Multi-Attribute Decision-Making (MADM)  
Alternatively, MADM methods are used to make preferential decisions over available alternatives or candidates and rank them according to a standard and predefined algorithm, regardless of the functional relationship between the attributes of each alternative or candidate [7]. This method generally involves two main steps: first, desired traits of attributes are selected from among those selected for study, and then the candidate materials are compared and ranked according to their degree of similarity to the desired traits. The candidate material with the closest similarity or distance to the selected desired traits is considered to be the optimal material for the given application [7].  
Under the MADM methodology, there are different philosophies that work with different approaches [33]. However, the popular method known as the technique for order preference by similarity to ideal solutions (TOPSIS), working with the reference level approach, is widely used for the material selection problem [14]. TOPSIS works based on the approach that positive and negative ideal solutions are created from a given set of data as reference levels and the optimum material is the one that lies closest to the positive ideal solution and furthest from the negative ideal solution, as illustrated in the Figure 3. The positive ideal solution is one that contains the best possible attributes from the given set of candidate materials; similarly, the negative ideal solution contains the worst possible attributes from the given set of candidate materials [14].  
The TOPSIS process is carried out as follows, according to the source [33]:  
Step 1:  
Create an evaluation matrix consisting of m alternatives and n criteria, with the intersection of each alternative and criteria given as \(x_{ij}\); we therefore have a matrix \(\left(x_{ij}\right)_{m\times n}\).  
Step 2:  
The matrix \(\left(x_{ij}\right)_{m\times n}\) is then normalized to form the matrix \(\left(R_{ij}\right)_{m\times n}=\left(r_{ij}\right)_{m\times n}\), using the linear normalization method:  
\[r_{ij}=\frac{x_{ij}}{\left(x_{ij}\right)max}\text{,if the criterion is positive, i =1,2,3,\ldots,m and j =1,2,3,\ldots,n,} \tag{4}\]  
or:  
\[r_{ij}=\frac{\left(x_{ij}\right)min}{x_{ij}}\text{,if the criterion is negative, i =1,2,3,\ldots,m and j =1,2,3,\ldots,n.} \tag{5}\]  
Step 3:  
Calculate the objective weighted normalized decision matrix using the equation:  
\[\left(T_{ij}\right)_{m\times n}=w\times\left(R_{ij}\right)_{m\times n}, \tag{6}\]  
Figure 3: Graphical description of TOPSIS method.  
where w is the weight factors for criteria.  
Step 4:  
Determine the worst alternative \(A_{w}\) and the best alternative \(A_{b}\):  
\[\begin{split}& A_{w}=\left[\min(t_{ij}|i=1,2,3,\ldots,m)\;\;for \;each\;j=1,2,3,\ldots,n\;\right],\\ & A_{b}=\left[\max(t_{ij}|i=1,2,3,\ldots,m)\;\;for\;each\;j=1,2,3, \ldots,n\;\right].\end{split} \tag{7}\]  
Step 5:  
Calculate the \(L^{2}\) distance between the target alternative \(i\) and the worst condition \(A_{w}\):  
\[d_{iw}=\sqrt{(\sum_{j=1}^{n}{(t_{ij}-t_{wj})^{2})}},\;\;i=1,2,3,\ldots,m, \tag{8}\]  
and the distance between the alternative \(i\) and the best condition \(A_{b}\):  
\[d_{ib}=\sqrt{(\sum_{j=1}^{n}{(t_{ij}-t_{bj})^{2})}},\;\;i=1,2,3,\ldots,m. \tag{9}\]  
Step 6:  
Calculate the similarity to the worst condition:  
\[s_{iw}=\frac{d_{iw}}{(d_{ib}+d_{iw})},\;0\leq s_{iw}\leq 1,\;\;i=1,2,3,\ldots,m, \tag{10}\]  
where \(s_{iw}=1\) if and only if the alternative solution has the best condition; and \(s_{iw}=0\) if and only if the alternative solution has the worst condition.  
Step 7:  
Rank the alternatives according to \(s_{iw}(i=1,2,3,\ldots,m)\).