## 2 Materials and Methods
### Description of the System
\begin{array}{c}if\beta(t)=1\ and\ s_{AR}(t)+U_{Ar}\leq S_{Ar} \ and\ s_{r}(t)<S_{Ar}\\ or\ if\beta(t)=1\ and\ S_{AR}-U_{Ar}\leq s_{Ar}(t)<S_{AR}\ and\ s_{r}(t)<S_{AR}-s_{Ar}(t)\\ \qquad\qquad\qquad\qquad\text{or}\ if\beta(t)=1\ and\ s_{Ar}(t)=S_{AR}ands_{r}(t)<d_{ AR}(t)\\ 0&if\beta(t)=0\ and\ s_{r}(t)=0\end{array} \right. \tag{14}\]  
In this section, we describe the methodology used to plan the production for manufactured and reconditioned products, in the period where the government sets a limitation for carbon emission on a random period (periodic mandatory carbon emission). First, we present the time horizon discretization for the studied problem (see Figure 2). The horizon \([0,T]\) is the finite horizon discretized into periods of time \(\Delta t\), where \(\Delta t\) is the simulation time step. The occurrence of a period under mandatory carbon emission in the system and the length of the period are generated by truncated normal distribution. * In order to determine the production quantity for manufacturing and reconditioning products under carbon regulation, we have provided a method to determine the quantity that respects the limited carbon emission and answers to the demand according to the determined production quantity under hedging point policy. We recall \(q_{pm}\) and \(q_{pr}\) present the carbon emissions of manufacturing and reconditioning respectively, and \(q_{l}\) is the limit quantity of carbon emission to not exceed. We have developed two algorithms (please see Algorithms 1 and 2 that are given in Appendices A and B) that build a function system capable to calculate the proportion of the production rate of the nearest manufactured and reconditioned products under the constraint to  
Figure 2: Studied time horizon. not exceed the limit of carbon emissions. The steps are as follows: We calculate the proportion of the production for new and reconditioned products:  
\[P_{SC}=\frac{U_{A}(t)}{U_{A}(t)+U_{Ar}(t)} \tag{15}\]  
\[P_{AC}=\frac{U_{Ar}(t)}{U_{A}(t)+U_{Ar}(t)} \tag{16}\]  
* We search, with two counters \(i\) and \(j\) running from 1 to the upper bound (\(u_{A}(t)\) and \(u_{Ar}(t)\)), to respect the constraint of not exceeding the limit of carbon emission: \[i.q_{pm}+j.q_{pr}<=q_{l}\] (17)
* We calculate the proportion: \[P_{m2 co}=\frac{i}{i+j}\] (18)  
\[P_{R2 CO}=\frac{j}{i+j} \tag{19}\]  
Then, we developed a function that gives the nearest values corresponding to the sum value \(P_{m2 co}\) and \(P_{R2 CO}\) closest to \(P_{SC}\) and \(P_{AC}\). The total profit function is determined by the difference between the total revenue and the total costs. The total revenue includes the total cost from the new products \(\sum\limits_{t=0}^{T}V_{A}(t)\cdot cv_{A}\), and the total cost for reconditioned products \(\sum\limits_{t=0}^{T}V_{Ar}(t)\cdot cv_{Ar}\). The total cost includes: the total production costs for new products \(\sum\limits_{T=0}^{T}u_{A}(t)\cdot cu_{A}\) and the total production costs for reconditioned products \(\sum\limits_{T=0}^{T}u_{Ar}(t)\cdot cu_{Ar}\), the total storage cost for new products \(\sum\limits_{T=0}^{T}s_{A}(t)\cdot cs_{A}\), the total storage cost for reconditioned products \(\sum\limits_{T=0}^{T}s_{Ar}(t)\cdot cs_{Ar}\), and the total storage cost for returned products \(\sum\limits_{T=0}^{T}s_{r}(t)\cdot cs_{r}\). The total return cost for worn products \(\sum\limits_{T=0}^{T}R(t)\cdot cr\), the total lost sales for new products \(\sum\limits_{T=0}^{T}P_{A}(t)\cdot cp_{A}\), and the total lost sales for reconditioned products \(\sum\limits_{T=0}^{T}P_{Ar}(t)\cdot cp_{Ar}\). The total amount of carbon emitted from producing new products over the horizon [0, \(T\)] is \(\sum\limits_{T=0}^{T}u_{A}(t)\cdot ct\cdot q_{pm}\) and from reconditioned products is \(\sum\limits_{T=0}^{T}u_{Ar}(t)\cdot ct\cdot q_{pr}\). We try to maximize the total profit function given by:  
\[F(t)=\sum\limits_{t=0}^{t=T}\left[