## 2 Materials and Methods
### Description of the System  
Figure 1 presents the model system, that is composed of two machines denoted \(M_{1}\) for manufacturing and \(M_{2}\) for reconditioning, which are subject to random repairs and failures. The machine \(M_{1}\) produces new products from raw material, and we supposed that it never starves. This assumption defines the general case of the manufacturing system where the raw materials are usually acquired. The machine produces new products at variable rate \(u_{A}(t)\) and they are stored in the stock \(S_{A}\). The variable \(u_{A}(t)\) represents the decided amount for producing new parts in the period \(t\). In fact, \(u_{A}(t)\) is determined at the period \(t\)-\(\Delta t\) according the machine state (\(\theta(t)\)) and the stock level of new parts \(s_{A}(t)\). The production rate \(u_{A}(t)\) takes a value between 0 and its maximum \(U_{A}\). The stock \(S_{A}\) satisfies the demand of new products \(d_{A}(t)\) coming from the Market 1. The sold new products will be used for a period \(\alpha\) and then will be collected and returned to the manufacturer, then, after, they are stored in the recovery inventory \(R\), to be reconditioned by the machine \(M_{2}\), and finally are stored in \(S_{Ar}\). The production rate of the reconditioning machine (\(M_{2}\)) is denoted \(u_{Ar}(t)\), which represents the decided amount for producing reconditioned parts in the period \(t\). Indeed, \(u_{Ar}(t)\) is determined at the period \(t\)-\(\Delta t\) according the machine state (\(\beta(t)\)), the stock level of reconditioned parts \(s_{Ar}(t)\) and the stock level of worn products \(s_{Ar}(t)\). The production rate \(u_{Ar}(t)\) takes a value between 0 and its maximum \(U_{Ar}\). The stock \(S_{Ar}\) satisfies the demand of reconditioned products \(d_{Ar}(t)\) coming from the Market 2. We supposed that the sold reconditioned products in Market 2 will be destroyed after their end of life. Indeed, in the real case, most reconditioned products are reused one time. In order to study the system under the carbon emission regulations used, we take into consideration the carbon emission emitted by producing the new/reconditioned products. In our model, the carbon emissions represent the quantity of carbon emitted from producing either new or reconditioned products. In other words, \begin{table}
\begin{tabular}{l l} \hline \(d_{Ar}(t)\) & demand for reconditioned products at time \(t\). \\ \(P_{A}(t)\) & quantity of unmet demand for new products at time \(t\). \\ \(P_{Ar}(t)\) & quantity of unmet demand for reconditioned products at time \(t\). \\ \(V_{A}(t)\) & quantity of new products sold at time \(t\). \\ \(V_{Ar}(t)\) & quantity of reconditioned products sold at time \(t\). \\ \(R(t)\) & quantity of worn products that are collected and returned from market 1 at time \(t\). \\ \(p\) & percentage of the returned products (worm products). \\ \(p^{*}\) & optimal percentage of the returned products. \\ \(\alpha\) & lifetime of products. \\ \(\theta(t)\) & state of the machine \(M_{1}\) at time \(t\). \\ \(\beta(t)\) & state of the machine \(M_{2}\) at time \(t\). \\ \(cv_{A}\) & unit selling price for new products. \\ \(cv_{Ar}\) & unit selling price for reconditioned products. \\ \(cu_{A}\) & unit production cost of new products. \\ \(cu_{Ar}\) & unit production cost of reconditioned products. \\ \(cs_{A}\) & unit storage cost for new products. \\ \(cs_{Ar}\) & unit storage cost for reconditioned products. \\ \(cr\) & unit storage cost for returned product. \\ \(cp_{A}\) & unit lost sales cost for new products. \\ \(cp_{Ar}\) & unit lost sales cost for reconditioned products. \\ \(ct\) & unit carbon emission cost. \\ \(q_{l}\) & quantity limit of carbon emission. \\ \(q_{pm}\) & carbon emission quantity of new products. \\ \(q_{pr}\) & carbon emission quantity of reconditioned products. \\ \(W\) & length of the mandatory carbon period. \\ \(F(t)\) & total profit function. \\ \(F(t)\)* & optimal value of total profit. \\ \hline \end{tabular}
\end{table} 
Table 1: Cont.