## 5 Numerical Example
### Sensitivity Study  
#### 5.4.1 Impact of Costs Variability  
\(C_{ps}=35\) and \(C_{c}=3550\)  
From Figures 7 and 8 and Tables 3 and 4, the bigger gap between the subcontractor unit cost and the cost of carbon penalty, the longer the subcontractor unit will take to produce and the larger the amount of carbon emitted during the production process. This result is predictable because the of the firm aim to minimize its total cost-function, which means that if the cost of penalty is bigger than the  
Figure 6: Optimal number of preventive maintenance actions.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \begin{tabular}{c} **Lot Produced by** \\ **Manufacturing** \\ **Unit** \\ \end{tabular} & \begin{tabular}{c} **Lot Produced by** \\ **Remanufacturing** \\ **Unit** \\ \end{tabular} & \begin{tabular}{c} **\(\alpha_{k}\)** \\ **Subcontracting** \\ **Unit** \\ \end{tabular} & \
\begin{tabular}{c} **\(\omega_{m}\)** \\ **Unit** \\ \end{tabular} &
\begin{tabular}{c} **\(w_{r}\)** \\ **Unit** \\ \end{tabular} \\ \hline
1 & 1563 & 0 & 1 & 670 & 271 & 0 \\
2 & 1754 & 0 & 1 & 438 & 435 & 0 \\
3 & 1713 & 87 & 1 & 428 & 768 & 894 \\
4 & 1751 & 47 & 0 & 13 & 597 & 1805 \\
5 & 2252 & 11 & 0 & 97 & 964 & 1887 \\
6 & 1988 & 17 & 0 & 53 & 909 & 1816 \\
7 & 825 & 119 & 1 & 1237 & 1120 & 1697 \\
8 & 349 & 97 & 1 & 1398 & 782 & 1599 \\
9 & 324 & 118 & 1 & 1294 & 436 & 1875 \\
10 & 187 & 55 & 1 & 1685 & 234 & 2256 \\
11 & 211 & 67 & 1 & 1895 & 310 & 2796 \\
12 & 799 & 54 & 1 & 1199 & 291 & 2742 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Production plan with \(C_{ps}=35\); \(C_{c}=3550\).  
cost of subcontracting, the firm will try to minimize the emission of carbon in order to minimize the total cost.  
\(C_{\text{ps}}=500\) and \(C_{c}=550\)  
We can also remark that the gap between the quantity of emitted carbon during the remanufacturing process with and without subcontracting remains almost the same in the final  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Period**} & **Lot Produced by** & **Lot Produced by** & \multicolumn{4}{c}{**Lot Produced by**} \\  & **Manufacturing** & **Remanufacturing** & \(\mathbf{\alpha_{k}}\) & **Subcontracting** & \(\mathbf{w_{m}}\) & \(\mathbf{w_{r}}\) \\  & **Unit** & **Unit** & **Unit** & **Unit** & **Unit** & \\ \hline
1 & 1600 & 0 & 1 & 686 & 391 & 0 \\
2 & 1160 & 0 & 1 & 774 & 253 & 0 \\
3 & 1172 & 100 & 1 & 781 & 215 & 279 \\
4 & 2083 & 84 & 0 & 27 & 458 & 786 \\
5 & 2274 & 30 & 0 & 46 & 798 & 710 \\
6 & 1838 & 48 & 0 & 48 & 773 & 606 \\
7 & 1151 & 106 & 1 & 768 & 819 & 1103 \\
8 & 1456 & 58 & 1 & 364 & 755 & 2026 \\
9 & 1197 & 54 & 1 & 798 & 896 & 2368 \\
10 & 1768 & 71 & 1 & 196 & 889 & 2491 \\
11 & 1798 & 52 & 1 & 200 & 845 & 2427 \\
12 & 1287 & 89 & 1 & 551 & 830 & 2951 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Production plan with \(C_{\text{ps}}=500\); \(C_{c}=550\).  
Figure 8: Amount of emitted carbon with and without subcontracting unit for \(C_{\text{ps}}=500\); \(C_{c}=550\). (a) Manufacturing unit; (b) Remanufacturing unit.  
Figure 7: Amount of emitted carbon with and without subcontracting unit for \(C_{\text{ps}}=35\); \(C_{c}=3550\). (a) Manufacturing unit; (b) Remanufacturing unit.  
period (around 2000 mu). This is mainly due to the priority order (the manufacturing unit has priority). In fact, the subcontractor will not take the maximal lot until one of our units has exceeded its allowed amount (which means that the indicator function corresponding to this unit will take 4). If both manufacturing and remanufacturing units exceed respectively \(Q_{cm}\) and \(Q_{rm}\), then the subcontractor will assist in the manufacturing. Which is why we remark in our case that the gap between \(Q_{r}\) with and without subcontracting remains almost the same in the final period when both units have exceeded their allowed amount.