## 6 Methodology
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Properties** & **L1** & **L2** & **L3** & **L4** & **D1** & **D2** & **D3** & **B1** & **B2** & **B3** \\ \hline CS, MPa & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 \\ TS, MPa & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 \\ DIS, \% & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 \\ Water absorption, \% & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 \\ Density, g/cm\({}^{3}\) & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 \\ Abrasion, \% & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 \\ P wave velocity, m/s & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 \\ Cost, LE & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 \\ Cost, LE & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Assignment of weights of to the tested rock samples. All methods were formulated according to the UBC method, which was developed and adjusted for application in this study. All equations were formulated and linked using cells in an Excel sheet that represented all properties, assuming that the property was considered in determining an alternative. Referencing to the value assigned to the \(j\)th criterion of the \(i\)th alternative, where \(x_{ij}\) is the decision matrix, the equivalent weight of the property is expressed as \(w_{1}\), \(w_{2}\),..., \(w_{n}\), and the TOPSIS processes are expressed in five steps using Equations (1)-(5). 1. Normalize the decision matrix. This can be calculated by \[\overline{X}_{ij}=\frac{X_{ij}}{\sqrt{\sum_{i=1}^{n}X_{ij}^{2}}}\] (1) \[r_{ij}=x_{ij\_m}=1\times 2k_{j},\,i=1,\,\ldots,\,m;\,j=1,\,\ldots,\,n,\,\, \text{where}\,\,r_{ij}\,\,\text{denotes the normalized value of the $j$th criterion for the $i$th alternative $Ai$.}\] 2. Calculate the weighted normalized decision matrix. \(V_{ij}\) can be calculated as \[V_{ij}=\overline{X}_{ij}\times W_{j}\] (2) \[v_{ij}=w_{j}\,\,r_{ij},\,i=1,\,\ldots,\,m;\,j=1,\,\ldots,\,n\,\,\text{(2)},\, \text{where}\,\,w_{j}\,\,\text{is the weight of the $j$th criterion or attribute.}\] 3. Determine the ideal positive and negative solutions \(S_{i}^{+}\) which can be calculated by \[S_{i}^{+}=\left[\sum_{j=1}^{m}\left(V_{ij}-V_{j}^{+}\right)^{2}\right]^{0.5}\] (3) 4. Calculate the Euclidean distance from the ideal worst condition. \(S_{i}^{-}\) can be described as \[S_{i}^{-}=\left[\sum_{j=1}^{m}\left(V_{ij}-V_{j}^{-}\right)^{2}\right]^{0.5}\] (4)  
Figure 3: Flowchart of the evaluation process. where \(V\!ij\) = weighted and normalized decision matrix (\(V\!ij\)), \(S^{+}\) = the Euclidean distance from the ideal best, \(S^{-}\) = the Euclidean distance from the ideal worst. 5. Calculate the performance score and ranking \(P_{i}\) This can be calculated by  
\[P_{i}=\frac{S_{i}^{-}}{S_{i}^{+}+S_{i}^{-}} \tag{5}\]  
Many researchers have studied the relationships between physio-mechanical properties and the behavior of limestone and other materials during general evaluations ([PERSON], 1992). Quality indices have been applied for different properties such as porosity, swelling, and compressive strength. The final results indicated that compressive strength decreased with increased porosity and swelling indices, and the swelling index increased with an increased porosity index ([PERSON] et al. (2009). [PERSON] et al. (2009) studied suitable materials for building, cement manufacture, and road pavement with respect to their physical and mechanical properties in the Sohag and Qena quarries near the Nile River; their results indicated quality indices of very good, good, fair, and poor for samples tested for different purposes ([PERSON] et al., 2009). Nigerian limestones were studied by [PERSON] (1991), who determined that the suitability of limestone in highway pavement depends on the index properties of strength, chemical properties, petrographic properties, and resistance to attrition. Recommendations were made for the use of selected limestones as pavement materials in Nigeria ([PERSON] et al. (1994).