## 2 Methodology
### Material Assessment
#### 2.1.2 Multi-Attribute Decision Making (MADM) Ranking  
In the second step, Multi-Attribute Decision Making (MADM) tools--Techniques for Order Preference by Similarity to Ideal Solutions (TOPSIS) and the Analytic Hierarchy Process (AHP)--are combined to rank the prescreened material candidates.  
Table 1 illustrates a typical MADM performance matrix composed of a series of alternatives, their attributes, and the weights of each attribute. \(X_{ij}\) is the rating of alternative \(i\) with respect to its attribute \(j\), while \(W_{j}\) is the weight of attribute \(j\) denoting its relative importance to the other attributes [28].  
In this study, the material candidates are regarded as the alternatives, and the attributes are the PCM's quantitative properties, including the latent heat of fusion, \(dH\) [k]-kg\({}^{-1}\)], specific heat, \(c_{\text{p}}\) [k]-kg\({}^{-1}\).K\({}^{-1}\)], thermal conductivity, \(k_{\text{PCM}}\) [W.m\({}^{-1}\).K\({}^{-1}\)], and the material density, \(\rho\) [kg.m\({}^{-3}\)]. For the weights, AHP in subjective weighting is employed to take the designers' preferences into consideration [26; 27].  
The TOPSIS algorithm is then used to seek the best material alternatives closest to the \"ideal solution.\" It comprises six steps [27; 30]:  
1. Construct the performance matrix \([X]_{mn}\) as displayed in Table 1;
2. Obtain a normalized matrix \([N]_{mn}\) of matrix \([X]_{mn}\) and multiply it with the weights obtained by the AHP method; \[N_{ij}=\frac{X_{ij}}{\sqrt{\sum_{i=1}^{m}X_{ij}{}^{2}}}\ \ \ i=1,\ 2,\ldots,\text{m};\ j=1,\ 2,\ldots,\text{n}\] (1) \[I_{ij}=N_{ij}\cdot W_{j}\] (2)
3. Determine the ideal solution and the negative ideal solution by \[I^{+}=\left\{\begin{array}{ccc}max&min\\ \left(I_{ij}/I\right)\,&\left(I_{ij}/I^{\prime}\right)\ \ /i=1,\ 2,\ldots,\text{m} \right\}=\left\{I_{1}^{+},\ I_{2}^{+},\ldots,I_{n}^{+}\right\}\] (3)  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Attribute 1** & **Attribute 2** & \(\ldots\) & **Attribute \(n\)** \\ \hline Material alternative 1 & \(X_{11}\) & \(X_{12}\) & \(\ldots\) & \(X_{1n}\) \\ Material alternative 2 & \(X_{21}\) & \(X_{22}\) & \(\ldots\) & \(X_{2n}\) \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ Material alternative m & \(X_{m1}\) & \(X_{m2}\) & \(\ldots\) & \(X_{mn}\\[I^{-}=\left\{\begin{array}{cc}min&max\\ \left(I_{ij}/I\right)\,&\left(I_{ij}/J^{\prime}\right)\ /i=1,\ 2,\ldots, \mathrm{m}\\ i&i\end{array}\right\}=\left\{I_{1}^{-},\ I_{2}^{-},\ldots,I_{n}^{-}\right\}\] (4) where \(J=(1,\,2\,\ldots\ n)/j\) is associated with material properties of positive impacts and \(J^{\prime}=(1,\,2\,\ldots\ n)/j\) of negative impacts;
4. Calculate the distances of each material alternative to the ideal and negative ideal solutions; \[D_{i}^{+}=\left\{\sum_{j=1}^{n}\left(I_{ij}-I_{j}^{+}\right)^{2}\right\}^{1/ 2}i=1,\,2,\ldots,\mathrm{m}\] (5) \[D_{i}^{-}=\left\{\sum_{j=1}^{n}\left(I_{ij}-I_{j}^{-}\right)^{2}\right\}^{1/ 2}i=1,\,2,\ldots,\mathrm{m}\] (6)
5. Measure the similarity of all alternatives to the worst solution (\(S=1\) for an ideal material and 0 for a worst possible material); \[S_{i}=\frac{D_{i}^{-}}{D_{i}^{+}+D_{i}^{-}}\] (7)
6. Rank all the similarities to obtain the best material alternatives.