## 3 Problem Formulation
### Production Policy
#### 3.1.2 Production Rates  
\(\bullet\) Manufacturing unit  
For the manufacturing unit, the production quantity can vary from period to another, which depends on the random demand and the inventory. Furthermore, since the time of maintenance is negligible compared to production time, the production rate denoted by \(P_{\text{m}}(k)\) can vary from the minimal production rate \(P_{mmin}\) to the maximal production rate of the machine \(P_{mMAX}\).  
\(\bullet\)   Remanufacturing unit  
Concerning the remanufacturing unit, the quantity of remanufacturing unit depends on the inventory level corresponding to products returned by customers. Unlike the case of manufacturing unit, the remanufacturing unit may not produce the total duration of the period if its production period overlaps the preventive maintenance period. In this case the preventive maintenance and production periods intersects in only one period. To identify this period, we introduce a binary variable \(b_{k}\) which takes 1 if the production period will overlap with a preventive action and 0 if else.  
\[b_{k}=\left\{\begin{array}{ll}1&if\ k=\lfloor h\times x\rfloor\ \forall\ h\in\left\{1,\ldots, \left\lfloor\frac{H}{x}\right\rfloor\right\}\\ 0&else\end{array}\right.\]  
The quantity to be reprocessed each period by the remanufacturing unit is represented as follows:  
\[P_{r}(k)=(1-b_{k})\times P_{r}(k)+b_{k}\times(1-y)\times P_{r}(k) \tag{3}\]  
with \(P_{r}(k)\) the quantity reprocessed when the production and preventive action does not overlap. Under these condition \(P_{r}(k)\) can vary from \(P_{min}\) to \(P_{rMAX}\).  
\(\bullet\)   Subcontractor UnitIn order to reduce the quantity of the emitted carbon by the manufacturing and remanufacturing unit, the firm can call on a subcontractor unit. In this work, this unit can only help either in the production or the reprocess of the returned products. For this, we introduce the binary variable \(\alpha_{k}\) which takes 1 if subcontractor unit will help in the manufacturing. The expression of lots produced by subcontractor unit will be expressed as follows:  
\[P_{s}(k)=\alpha_{k}\times P_{s}(k)+(1-\alpha_{k})\times P_{s}(k) \tag{4}\]  
The question that may arise here is: under what condition will the subcontractor assist in manufacturing, not in remanufacturing? To set this condition, we will define two functions by interval depending on the amount of emitted carbon by manufacturing and remanufacturing units. We will assign a number from 1 to 4 to four different zones proportional to the quantity of emitted carbon. These functions will be expressed as follows:  
\[f_{m}(Q_{m})=\left\{\begin{array}{rl}1\;if\;Q_{1m}\leq Q_{m}\leq Q_{mMAX}\\ 2\;if\;Q_{2m}\leq Q_{m}\leq Q_{1m}\\ 3\;if\;0\leq Q_{m}\leq Q_{2m}\\ 4\;if\;Q_{m}\leq 0\end{array}\right.\]  
\[f_{r}(Q_{r})=\left\{\begin{array}{rl}1\;if\;Q_{1r}\leq Q_{r}\leq Q_{rMAX}\\ 2\;if\;Q_{2r}\leq Q_{r}\leq Q_{1r}\\ 3\;if\;0\leq Q_{r}\leq Q_{2r}\\ 4\;if\;Q_{r}\leq 0\end{array}\right.\]  
The quantities of carbon defining intervals (\(Q_{1m}\), \(Q_{2m}\), \(Q_{2r}\),...) are set by the firm depending on the maximal amount permitted by government (Figure 3). If \(f_{m}<f_{r}\) means that remanufacturing unit is closer to the maximum allowed quantity, the subcontractor will then assist the remanufacturing unit in the reprocess of returned products. Since the pressing of the manufacturing machine is more compared to the remanufacturing unit, we also assume that assistance of manufacturing unit has priority. This means that if \(f_{m}=f_{r}\) subcontractor unit will assist manufacturing unit. Under these assumptions, the function ensuring the switching of subcontractor from a unit to another is:  
\[\alpha_{k}=\left\{\begin{array}{rl}1\;if\;f_{m}\geq f_{r}\\ 0\;else\end{array}\right.\]  
Figure 3: Dashboard functions. (**a**) Remanufacturing Unit; (**b**) Manufacturing Unit.