## 3 Description of the MRP Mathematical Model with Remanufacturing Parts
### Disassembly-Assembly Plan  
The mathematical model proposed below leads to minimize these costs: inventory holding costs of different components (disassembled components, leaf-items, new components ordered from external resources, assembled components at level 1, finished product), remanufacturing costs, disposal costs, purchasing costs of new components and of used/end-of-life product, and backlogging cost of finished product. This mathematical function optimizes the order date of used/end-of-life product as well as the release dates of new components at level 2 over a finite horizon. The mathematical expressions mentioned below are used to clarify the analytical proposed model:  
The disassembly date \(Td_{i,1}\) for each component \(C_{i,1}\) is issued from the disassembly of the used/end-of-life product. This date represents the initiation of the disassembly process of \(C_{i,1}\). The disassembly process of each component begins after a random lead-time that includes the  
Figure 4: Description of one manufacturing finished product planning with a maintenance plan. component transmission, component preparation, etc. If the component is ready to be disassembled and the system is occupied by another operation, the component is stocked until the system is released. \[Td_{i,1}=T_{PR}+L_{PR}+D_{PR}+L_{i,1}+Str_{i,1} \tag{1}\]  
We note by \(Str_{i,1}\) the stock period of each component is \(C_{i,1}\) which is calculated as follows:  
We suppose that:  
\(z_{k}=1\): If the machine is occupied in \([k\cdot\Delta t;\ (k+1)\cdot\Delta t]\). \(z_{k}=0\): If the machine is available in \([k\cdot\Delta t;\ (k+1)\cdot\Delta t]\). ```
1:\(Str_{i,1}=0\);
2:\(\Delta t=1\);
3:\(k=T_{PR}+L_{PR}+D_{PR}+L_{i,1}\);
4:\(while(z_{k}=1)\) do
5:\(Str_{i,1}=Str_{i,1}+\Delta t\);
6:\(k++\);
7:end do. 8:end. ```  
The assembly date for the finished product is \(I_{PF}\): the assembly of the finished product is started when all components of level 1 are ready:  
\[I_{PF}=\underset{i=1\dots N_{1}}{max}(I_{i,1}+A_{i,1}+L_{i,1}) \tag{3}\]The maximum between the end of the finished product assembly and the planned delivery time is noted by \(\left(I_{PF}+A_{PF}\right)^{+}\):  
\[\left(I_{PF}+A_{PF}\right)^{+}=max(I_{PF}+A_{PF},T_{PF}) \tag{4}\]  
The minimum between the end of the finished product assembly and the planned delivery time is noted by \(\left(I_{PF}+A_{PF}\right)^{-}\):  
\[\left(I_{PF}+A_{PF}\right)^{-}=max(I_{PF}+A_{PF},T_{PF}) \tag{5}\]  
The objective function includes:  
The inventory costs of disassembled components:  
\[C_{d}=\sum\limits_{i=1}^{N_{1}}h_{i,1}(Td_{i,1}-(T_{PR}+D_{PR}+Lr_{i,1}+L_{PR})) \tag{6}\]  
The inventory costs of components at level 2:  
\[C_{sc2}=\sum\limits_{i=1}^{N_{1}}\sum\limits_{j=1}^{S_{i,1}}h_{j,2}(I_{i,1}-( Td_{i,1}+D_{i,1}+L_{j,2}))+\sum\limits_{i=1}^{N_{1}}\sum\limits_{j=1}^{E_{i,1}}h_{j,2}(I_{i,1 }-(X_{j,2}+L_{j,2})) \tag{7}\]  
The inventory costs of components at level 1:  
\[C_{sc1}=\sum\limits_{i=1}^{N_{1}}\left(h_{i,1}\times(I_{PF}-(I_{i,1}+L_{i,1}+ A_{i,1}))\right) \tag{8}\]  
Backlogging costs for the finished product:  
\[C_{bPF}=b\times((I_{PF}+A_{PF})^{+}-T_{PF}) \tag{9}\]  
Inventory holding costs for the finished product:  
\[C_{rPF}=r\times(T_{PF}-(I_{PF}+A_{PF})^{-}) \tag{10}\]  
Therefore, the total function cost denoted by \(C(T_{PR},X)\) which is composed of inventory holding costs of the different components (disassembled components issued from the disassembly of the used/end-of-life product, components at level 1 and 2, and the finished product), backlogging costs, purchasing costs (purchasing of the used/end-of-life product and ordered components of level 2 from external resources), renorming, and disposal costs. \[