## 3 Materials and Methods
### Uncertainty Propagation
\begin{split} NWC_{t}^{scen}=\frac{R_{T}^{scen}}{\text{CTR}}+\frac{R_{T} ^{scen}}{\text{DTR}}+\frac{T_{T}^{scen}-\sum_{prad}DA_{t}^{prad}}{\text{ITO}}\\ -\frac{T_{t}^{scen}-\sum_{prad}DA_{t}^{prad}}{\text{DTP}},scen\in\{ base,inv\}\\ RV_{T}^{base}=\frac{R_{T}^{scen}}{\text{CTR}}+0.7\frac{R_{T} ^{scen}}{\text{DTR}}+0.7\frac{T_{T}^{scen}-\sum_{prad}DA_{T}^{prad}}{\text{ITO}} \\ -\frac{T_{T}^{scen}-\sum_{prad}DA_{t}^{prad}}{\text{DTP}},scen\in\{ base,inv\},\\ \end{split}  \tag{31}\]

The notation used in model are presented in Table 1.

The Net Present Value of the proposed OC-sheet line is obtained with Equation (18), in which the incremental cash flows are defined by Equation (19) and decomposed by Equations (20)-(29) into operating profit after tax, depreciation, variable input costs and fixed overheads. The annual variation in net working capital is captured by Equation (30), while the stock of working capital itself is evaluated with the revised Equation (31), which relates cash, inventories, receivables and payables to four turnover ratios (CTR, ITO, DTR and DTP). At the end of the planning horizon, the model releases tied-up capital according to the Wilcox convention [5] implemented in the revised Equation (32), i.e., it recovers 100% of cash, 70% of the book value of inventories and receivables, and the full 100% of outstanding payables. \begin{table}
\begin{tabular}{l} \hline \hline
**Indices** \\ \hline \(t\)—investment period: \(\{1,\ldots,\text{T}\}\); \\ _scen_—scenario: _base_—baseline, _inv_—investment; \\ _prod_—product: OC—OC sheet, _HDG_—HDG sheet, _CR_—CR sheet, _scrap_—scrap; \\ \hline \hline \multicolumn{2}{l}{**Input parameters**} \\ \hline _TA_—tax, \\ _CAP_\({}^{prod}\)_—capacity of _prod_ plan; \\ _MS_\({}^{prod}\)_—market share of _prod_ product; \\ _DTR_—receivable turnover ratio; \\ _CTR_—cash turnover ratio; \\ _ITO_—inventory turnover ratio; \\ _DTP_—payable turnover ratio; \\ _PUC_\({}^{prod}\)_—per-unit consumption of sheet required to produce _prod_ sheet; \\ _FC_\({}^{prod}\)_—fixed cost of _prod_ plan; \\ _VC_\({}_{O}^{prod}\)_—other annual variable production costs per ton for _prod_ sheet; \\ Auxiliary variables \\ _PAT_\({}_{t}^{scen}\)_—profit after tax in _scen_ scenario in \(t\); \\ _DA_\({}^{prod}\)_—annual depreciation of _prod_ plan in t; \\ _A_\({}_{W}\)_\({}_{C}^{scen}\)_—the change in net working capital in _scen_ scenario in \(t\); \\ _RV_\({}_{T}^{scen}\)_—residual value in _scen_ scenario in \(T\); \\ _R_\({}_{T}^{scen}\)_—revenue in _scen_ scenario in \(t\); \\ _TC_\({}_{T}^{scen}\)_—total cost in _scen_ scenario in \(t\); \\ _SV_\({}_{t}^{prod,scen}\)_—sales volume of _prod_ product in _scen_ scenario in \(t\); \\ _N_W_\({}_{C}^{scen}\)_—networking capital in year \(t\) in scenario _scen_; \\ _VC_\({}_{RM}\)_\({}_{T}^{prod}\)_—cost of CR sheet per ton of _prod_ sheet in year t; \\ _SP_\({}_{t}^{prod}\)_—sales forecast for _prod_ product; \\ \hline \multicolumn{2}{l}{**Uncertain variables**} \\ \hline \(P_{t}^{prod}\)_—price of _prod_ per ton in year \(t\); \\ _ACP_\({}_{t}^{prod}\)_—apparent consumption forecast for _prod_ product in \(t\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation for parameters and variables used in NPV calculation. Prices and apparent consumption for the analyzed products are inter-related rather than independent, displaying complex patterns of correlation and mutual dependence. Two relationship types can be distinguished: correlations and reciprocal dependencies. The pair-wise price correlations are summarized in Table 2, and the correlation coefficient between the apparent consumption of HDG sheet and OC sheet is 0.532.