## 3 Materials and Methods
### Uncertainty Propagation
Interval-regression coefficients that characterize the interdependencies among the drift parameters (\(\mu_{\text{i}}\)) of the individual price series are provided in Table 3; the corresponding coefficients for apparent consumption appear in Table 4. Interval-regression coefficients (\(a1\), \(a2\)) were computed by solving the pair of inequalities given in Equations (6) and (7), following [PERSON] & Cerny's method. For a steel plant operating with the leverage typical of the industry, we assume a real weighted-average cost of capital (WACC) of 10% p.a. and a real risk-free rate of 5% p.a. [4]. Polish historical datasets (1996-2022, constant prices-2022 USD) show that, in real terms, OC-sheet prices increased at an average annual rate of 0.89%. Over the same period, average annual growth rates were 1.46% for HDG sheet, 1.73% for CR sheet, and 1.63% for \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Independent Variable**} & \multicolumn{3}{c}{**Dependent Variable**} \\ \cline{3-4}  & & **HDG Sheet** & **OC Sheet** \\ \hline HDG sheet & \(a_{1}\) & & [\(-\)0.239; 0.800] \\  & \(a_{2}\) & & [\(-\)0.050; 0.167] \\ \hline OC sheet & \(a_{1}\) & [\(-\)1.589; 3.461] \\  & \(a_{2}\) & [\(-\)0.040; 0.088] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The interval-regression coefficient capturing the interdependence between drift parameters of apparent consumption for the HDG sheet and OC sheet. Each bracket shows [lower; upper] the bounds derived from Equations (6) and (7), respectively; \(a_{1}\) is the slope, \(a_{2}\) the intercept. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Scrap** & **CR Sheet** & **HDG Sheet** & **OC Sheet** \\ \hline Scrap & 1.000 & 0.955 & 0.892 & 0.853 \\ CR sheet & 0.955 & 1.000 & 0.961 & 0.921 \\ HDG sheet & 0.892 & 0.961 & 1.000 & 0.911 \\ OC sheet & 0.853 & 0.921 & 0.911 & 1.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation matrix for product prices. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Independent Variable**} & \multicolumn{3}{c}{**Dependent Variable**} \\ \cline{3-5}  & **Scrap** & **CR Sheet** & **HDG Sheet** & **OC Sheet** \\ \hline \multirow{2}{*}{_Scrap_} & \(a_{1}\) & & [\(-\)0.060; 1.943] & [0.369; 1.378] & [0.589; 1.554] \\  & \(a_{2}\) & & [\(-\)0.018; 0.001] & [\(-\)0.006; \(-\)0.002] & [0.003; 0.0098] \\ \hline \multirow{2}{*}{_CR sheet_} & \(a_{1}\) & [\(-\)0.897; 2.914] & & [\(-\)1.611; 3.287] & [\(-\)1.689; 3.951] \\  & \(a_{2}\) & [\(-\)0.013; 0.041] & & [\(-\)0.015; 0.039] & [\(-\)0.031; 0.069] \\ \hline \multirow{2}{*}{_HDG sheet_} & \(a_{1}\) & [0.721; 1.489] & [0.399; 1.598] & & [0.531; 1.811] \\  & \(a_{2}\) & [0.003; 0.006] & [\(-\)0.011; \(-\)0.003] & & [0.004; 0.016] \\ \hline \multirow{2}{*}{_OC sheet_} & \(a_{1}\) & [0.063; 1.689 & [\(-\)2.160; 3.866] & [\(-\)0.599; 2.112] & \\  & \(a_{2}\) & [\(-\)0.012; 0.003] & [\(-\)0.062; 0.040] & [\(-\)0.022; 0.007] & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Interval-regression coefficient capturing the interdependence between drift parameters of the individual product-price series. Each bracket shows [lower; upper] the bounds derived from Equations (6) and (7), respectively; \(a_{1}\) is the slope, \(a_{2}\) the intercept. scrap. To generalize these trends we represent the common drift term across all products by the trapezoidal fuzzy number (0.85%, 1.10%, 1.30%, 1.50%), reflecting the structural shifts observed in the market--particularly during the 2006-2007 economic boom. A review of apparent-consumption data for 1996-2022 shows that demand for HDG sheet and OC sheet grew on average by 7.55% and 8.65% per year, respectively. To generalize these dynamics, we adopt a common drift term for all products, expressed as the trapezoidal fuzzy number (6.0%, 6.5%, 7.0%, 7.5%), which captures the structural changes the sector underwent--particularly in the boom years of 2006-2007. Volatility is estimated as the standard deviation of log-returns over 2000-2022.