## Appendix A  
**Proof. ** It is assumed that the demand variable has its first and second statistic moments perfectly known for each period \(k\), that is, \(E\{D(k)\}=\hat{D}(k)\) and \(V_{D(k)}=\sigma^{2}\) for each \(k\). The inventory variables \(w_{m}(k)\) and \(w_{r}(k)\) are statistically described respectively by their means \(E\{w_{m}(k)\}=\hat{w}_{m}(k)\) and \(E\{w_{r}(k)\}=\hat{w}_{r}(k)\) as well as their variances \(V_{w_{m}(k)}=E\Big{\{}\big{(}w_{m}(k)-\hat{w}_{m}(k)\big{)}^{2}\Big{\}}\) and \(V_{w_{r}(k)}=E\Big{\{}\big{(}w_{r}(k)-\hat{w}_{r}(k)\big{)}^{2}\Big{\}}\). The inventory balance of manufacturing unit can be reformulated as:  
\[\hat{w}_{m}(k+1)=\hat{w}_{m}(k)+\hat{P}_{MT}(k)+\hat{P}_{RT}(k)-\hat{D}(k) \tag{10}\]  
Likewise, the inventory balance of remanufacturing unit can be reformulated as:  
\[\hat{w}_{r}(k+1)=\hat{w}_{r}(k)+\beta\times\hat{D}(k-\rho)\times(1-\hat{\varphi }_{k})-\hat{P}_{RT}(k) \tag{11}\]  
\[k=\{1,2,\,\ldots,\,H-1\}\text{ and }k\geq\tau\]  
With  
\[\hat{P}_{MT}(k)=\big{[}\hat{P}_{m}(k)+\alpha_{k}\times\hat{P}_{S}(k)\big{]}\]  
\[\hat{P}_{RT}(k)=\big{[}\big{(}(1-b_{k})\times\hat{P}_{r}(k)+b_{k}\times(1-y) \times\hat{P}_{r}(k)\big{)}+(1-\alpha_{k})\times\hat{P}_{S}(k)\big{]}\]  
* If we take the difference between Equations (5) and (10) we obtain:  
\[ \begin{array}{ll}w_{m}(k+1)-\hat{w}_{m}(k+1)&=(w_{m}(k)-\hat{w}_{m}(k))\\ &+\big{(}\underbrace{P_{MT}(k)-\hat{P}_{MT}(k)}_{0}\big{)}+\underbrace{(P_{RT}( k)-\hat{P}_{RT}(k))}_{0}-\big{(}D(k)-\hat{D}(k)\big{)}\end{array} \]  
\[ \begin{array}{ll}E\Big{\{}\big{(}w_{m}(k+1)-\hat{w}_{m}(k+1)\big{)}^{2}\Big{\}} \\ =E\Big{\{}\big{(}w_{m}(k)-\hat{w}_{m}(k)\big{)}^{2}\Big{\}}-E\Big{\{}\big{(}D( k)-\hat{D}(k)\big{)}^{2}\Big{\}}\\ -2\cdot E\big{\{}\big{(}(w_{m}(k)-\hat{w}_{m}(k))\cdot\big{(}D(k)-\hat{D}(k) \big{)}\big{\}}\end{array} \]  
Since \(w_{m}(k)\) and \(D(k)\) are independent random variables, we can deduce that:  
\[ \begin{array}{ll}E\big{\{}((w_{m}(k)-\hat{w}_{m}(k))\cdot\big{(}D(k)-\hat{D }(k)\big{)}\big{\}}\\ =E\big{\{}((w_{m}(k)-\hat{w}_{m}(k))\big{\}}\times E\big{\{}\big{(}D(k)-\hat{D }(k)\big{)}\big{\}}\end{array} \]  
Hence:  
\[ \begin{array}{ll}E\big{\{}((w_{m}(k)-\hat{w}_{m}(k))\big{\}}=E\{w_{m}(k)-E \{\hat{w}_{m}(k)\}=0\\ E\big{\{}\big{(}D(k)-\hat{D}(k)\big{\}}\big{)}=E\{D(k)\}-E\{\hat{D}(k)\}=0 \end{array} \]  
Therefore  
\[\Big{\{}(w_{m}(k+1)-\hat{w}_{m}(k+1))^{2}\Big{\}}=E\Big{\{}(w_{m}(k)-\hat{w} _{m}(k))^{2}\Big{\}}-E\Big{\{}\big{(}D(k)-\hat{D}(k)\big{)}^{2}\Big{\}}\]  
If we assume that \(V_{w_{m}}(k=0)=0\) and that \(\sigma_{D}\) is constant and equal to \(\sigma\) for all periods, we can deduce that \(V_{w_{m}}(k+1)=V_{w_{m}}(k)+\sigma^{2}\)\(k\geq 1\). For \(k=0\), \(V_{w_{m}}(1)=\sigma^{2}\), \(V_{w_{m}}(k)=k\times\sigma^{2}\). Since \(V_{w_{m}}(k)=E\Big{\{}\big{(}w_{m}(k)-\hat{w}_{m}(k)\big{)}^{2}\Big{\}}=E\Big{\{} w_{m}(k)^{2}\Big{\}}-\hat{w}_{m}(k)^{2}\), we can write  
\[E\Big{\{}w_{m}(k)^{2}\Big{\}}-\hat{w}_{m}(k)^{2}=k\times\sigma^{2}.\]  
Hence,  
\[E\Big{\{}w_{m}(k)^{2}\Big{\}}=k\times\sigma^{2}+\hat{w}_{m}(k)^{2} \tag{12}\]  
* If we take the difference between Equations (6) and (11) we obtain: \[ \begin{array}{ll}w_{r}(k+1)-\hat{w}_{r}(k+1)\\ =(w_{r}(k)-\hat{w}_{r}(k))+\beta\times\big{(}D\big{(}k-\frac{\rho}{\Delta t} \big{)}-\hat{D}\big{(}k-\frac{\rho}{\Delta t}\big{)}\big{)}\times(1-\hat{ \varphi}_{k})\\ -\big{(}P_{RT}(k)-\hat{P}_{RT}(k)\big{)}\end{array} \] \[\begin{array}{ll}k=\{1,2\ldots,H-1\}\text{ and }k\geq\frac{\rho}{ \Delta t}\]  
Our production and maintenance problem will then be defined as follows:  
\[Min\;C_{T}=Min\left\{