## 4 Cost Function to Be Minimized
### Approach for the Quantitative Discounting of Selected Cost Indexes  
Readers may observe that certain terms in Equation (14) are denoted with the superscript %. This graphic notation distinguishes the capital costs for which the authors have decided to apply quantitative discounting. Indeed, it is reasonable to assume that by purchasing large quantities of certain materials or components, as might occur when designing long lines, the producer or investor can benefit from an appropriately discounted unit price compared with that when purchasing smaller quantities.  
Listed here are the logical steps for calculating this quantitative discount for each of the cost indexes marked with the superscript % in Equation (14):  
1. A reference quantity (called \(q_{ref_{x}}\), where the subscript \(x\) can be replaced with the corresponding cost term) is set for each discounted index, whether it be a length, a weight, a volume, or the number of components. These quantities correspond to the reference case for which the authors have assumed fixed unit costs (\(c_{x}\)), such as the materials required for a 50 km HTS cable. By carrying out parametric analyses with the proposed tool, different case studies from the reference one can be investigated, resulting in different optimized designs and material or component quantities. If these quantities are greater than their reference values, then a reduction in the unit cost is expected. Conversely, if the optimized quantities fall below their reference values, then it is reasonable that the buyer will have to pay a higher unit price.
2. A multiplicative coefficient relative to the reference quantity is set (referred to as \(m_{\%_{x}}\)), corresponding to a discount of the unit cost, called \(d_{\%_{x}}\). To give a quantitative example, supposing \(q_{ref_{x}}\) is equal to 100 km, \(m_{\%_{x}}\) is equal to 5, and \(d_{\%_{x}}\) is equal to 20%, if the optimized cable design requires purchasing a quantity of the material or component \(x\) equal to \(q_{ref_{x}}\). \(m_{\%_{x}}\)= 500 km, then the corresponding unit cost of that material or component will be lowered by 20% compared with the one set for the reference quantity. The parameter \(m_{\%_{x}}\) is also used as a divisor of the reference quantity, resulting in an increase in the unit cost when the optimized quantity is lower than the reference one, corresponding again to \(d_{\%_{x}}\).
3. It is assumed that the discount of the unit cost is linearly proportional to the optimized quantity. This discounting process is applied within maximum and minimum discounting rates compared with the reference unit cost. This corresponds to assuming that the buyer's or manufacturer's leverage on the unit price is limited to a certain range. For simplicity, both thresholds are set equal to \(d_{max\%_{x}}\), which might then be positive or negative. In mathematical form, the above can be summarized as \[c_{x}{}^{\%_{x}}=c_{x}\left(1-\frac{d_{\%_{x}}}{m_{\%_{x}}-1}\right)\left( \frac{q_{x}-q_{ref_{x}}}{q_{ref_{x}}}\right)\] (15) where \(c_{x}\) and \(c_{x}{}^{\%_{x}}\) are the unit cost of the material or component \(x\) before and after the quantitative discounting process, respectively, and \(q_{x}\) is the quantity of the material or component \(x\) computed during the optimization.  
Figure 5 illustrates the described steps by showing the correlation between the unit price of the generic material or component \(x\) and its quantity.  
In the formulae presented in the next sections which refer to the cost indexes outlined in Equation (14), the calculation for the discount of the unit cost of the corresponding material or component presented here will be omitted to avoid overcomplication. However, the unit costs will be marked with the superscript %, and in Section 7.1, examples of the \(q_{ref_{x}}\), \(m_{\%_{x}}\), \(d_{\%_{x}}\), and \(d_{max\%_{x}}\) values for the individual indexes will be provided.