## 4 Research Methodology
### Life-Cycle Cost Analysis  
To explore the financial efficiency of the model, the authors performed a LCCA comparing the \"as-is\" constant price ratio versus the use of the ore purchase index model (Equation (6)). The economic analysis was performed for a best (high-grade ore), worse (mid-grade ore), and worst (low-grade ore) case of Ni and Fe content. These cases are shown in Table 3.  
The authors performed a LCCA for these two cases, using a Monte Carlo simulation for data analysis. The three cases in Table 3 are used to calculate the operational profit from the manufacturing costs and revenue. These are calculated using cost Models 1, 2, or 3 as applicable and Company A's associated manufacturing costs, sales, and revenues represented in Figure 7. After the operational profits are calculated, uncertainty is incorporated using a triangular distribution and standard deviation from the best, worse, and worst cases, as shown in Equation (9) and (10) [43]:  
\[t_{e}=\frac{t_{o}+4t_{m}+t_{p}}{6} \tag{9}\]  
where \(t_{e}\) denotes the weighted average of expected profit (in millions of US dollars, M USD); \(t_{o}\) denotes the best expected profit (M USD); \(t_{m}\) denotes the worse expected profit (M USD); and \(t_{p}\) denotes the worst expected profit (M USD).  
\[\sigma=\frac{t_{p}-t_{o}}{6} \tag{10}\]  
where \(\sigma\) denotes the standard deviation (M USD); \(t_{o}\) denotes the best expected profit (in percentage); and \(t_{p}\) denotes the worst expected profit (M USD).  
The investment cost, operational cost, operational sales, depreciation, and taxes are all assumed and included in the LCCA. Table 4 shows the variables used for the LCCA, as experienced by Company A.  
From these variables, an LCCA analysis was performed with and without using the model under their three cases. Utilizing LCCA calculation best practices [44; 45; 46; 47; 48; 49; 50], the NPV, internal rate of return (IRR), and profitability index (PI) are calculated using the formulas below:  
\begin{table}
\begin{tabular}{l l l} \hline \hline  & & **Condition** \\ \hline \multirow{3}{*}{Not-Applying Ore Purchase Index Model} & Case 1 (Best) & Ni content 2.0\%, Fe content 15\% in an Ore \\  & Case 2 (Worse) & Ni content 1.9\%, Fe content 17\% in an Ore \\  & Case 3 (Worse) & Ni content 1.8\%, Fe content 19\% in an Ore \\  & Case 1 (Best) & Ni content 2.0\%, Fe content 15\% in an Ore \\ Applying Ore Purchase Index Model & Case 2 (Worse) & Ni content 1.9\%, Fe content 17\% in an Ore \\  & Case 3 (Worse) & Ni content 1.8\%, Fe content 19\% in an Ore \\ \hline \hline \end{tabular}
\end{table}
Table 3: With and without ore purchase index model LCCA: best, worse, and worst cases.  
\[\text{NPV}=\sum_{t=1}^{T}\frac{C_{t}}{(1+r)^{t}}-C_{0} \tag{11}\]  
where _NPV_ denotes the net present value (_M USD_); \(t\) denotes number of time periods (in years from 1 to \"T\" years); C\({}_{t}\) denotes net cash inflow during period \(t\) (M USD); _C\({}_{0}\)_ denotes initial investment costs (M USD); and \(r\) denotes the discount rate (in _percentage_).  
\[IRR=r\;when\;NPV=0 \tag{12}\]  
where _IRR_ denotes the internal rate of return (in percentage); _NPV_ denotes Equation (11); and \(r\) denotes the discount rate (in percentage).  
\[PI=\frac{NPV+Initial\;Investment}{Initial\;Investment} \tag{13}\]  
where _PI_ denotes the profitability index (in percentage); _NPV_ denotes the net present value (M USD); and _Initial Investment_ denotes the initial investment costs (M USD).  
These calculations were performed using the Excel @Risk Monte Carlo analysis for both the case of not using a model and using a model. The findings are presented as distribution curves depicting the likelihoods of certain NPV, IRR, and PIs occurring.