## 5 Numerical Example
### Sensitivity Study
#### 5.4.2 Impact of the Variability of the Proportion of emitted Carbon  
From Tables 5 and 6, we can remark that the variability of the proportion of emitted carbon to produce an item has an impact on the decision variable \(\alpha_{k}\) which indicates whether the subcontractor assists in the manufacturing or in the remanufacturing. In fact, by increasing the proportion of emitted carbon while maintaining the maximal allowed amount constant, both units will exceed this quantity earlier, which means that their indicator functions will take 4 and due to the priority order the subcontractor will assist in the manufacturing. By decreasing the value of \(\xi\), the remanufacturing unit will exceed its allowed amount earlier than the manufacturing unit because \(Q_{cm}\) is greater than \(Q_{rm}\). The indicator function of the remanufacturing is likely to take a bigger value than the one corresponding to the manufacturing unit. This means that the subcontractor unit will help the manufacturing unit in order to reduce its emission of carbon. The approach proposes a new joint production and maintenance policy, which considers an environmental aspect for a forecasting problem of a reverse logistic system subject to a random degradation. The satisfaction of a random demand under a given service level, and at the same time respecting the environmental constraint characterized by the harmful emissions to the environment, may be sanctioned by an environmental tax, and requires that the company has found new strategies to meet these different objectives. The objective of this study is to propose an environmental production and maintenance policy optimization by proposing the subcontractor as a solution in order to satisfy the random demand and decrease the carbon tax. In this context, an economical production plans for the manufacturing, remanufacturing and subcontract units as well as an optimal maintenance strategy by developing a correlation between the production and maintenance. Firstly, we have integrated the subcontractor solution on the production policy under a new switching approach which determines when the subcontractor can help either manufacturing or remanufacturing a unit during the production according to the dashboard, which defines different sectors of the quantity of emitted carbon in order to minimize the total cost of production, inventory and carbon tax. Secondly, using a sequential approach by integrating the economical production plan of manufacturing unit obtained by the production policy in the maintenance strategy, we determine the optimal number of preventive maintenance actions. The originality of this study in maintenance terms is to consider the influence of the production variation on the degradation degree of the manufacturing unit by establishing a correlation between the production and maintenance characterized by an analytical equation that shows the failure rate progress of the manufacturing unit according to its use, respecting at the same time the continuity of the reliability of the manufacturing unit from one period to another. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Period** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline \(\alpha_{1}\) & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ Q\({}_{m}\) without & 235,886 & 222,129 & 208,590 & 194,623 & 180,637 & 170,746 & 160,861 & 146,529 & 134,697 & 120,617 & 108,011 & 96,545 \\ Q\({}_{m}\) & 238,471 & 227,704 & 219,584 & 205,617 & 191,631 & 181,740 & 171,785 & 157,522 & 145,691 & 131,610 & 119,004 & 107,538 \\ Q\({}_{m}\) without & 2000 & 2000 & 1654 & 1105 & 438 & -38 & -822 & -165 & -206 & -2706 & -3286 & -3985 \\ Q\({}_{m}\) & 2000 & 2000 & 1654 & 1471 & 1068 & 929 & 816 & 995 & 475 & 274 & 186 & 116 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Amount of emitted carbon during the different process with \(Q_{cm}\) = 200,000 and \(Q_{cr}\) = 2000.