## 2 Bi-Objective Model for the Design and Management of RMSs
### Optimization Model Formulation
(2017):  
\[\begin{array}{ll}\sum\limits_{r=1}^{R}MP_{pt}=1&\forall t,\ p\\ \sum\limits_{n:B_{nm}=1}^{N}L_{rptt}=MP_{pt}\cdot\sum\limits_{\alpha_{m}^{R}=0} G_{orm}&\forall\ p,\ t,\ r,\ m\\ Z_{nt}\leq\sum\limits_{p=1}^{P}L_{rptt}&\forall\ r,\ n,\ t\\ L_{rptt}\leq Z_{nrt}&\forall\ p,\ r,\ n,\ t\\ \sum\limits_{r=1}^{R}Z_{nrt}\leq 1&\forall\ n,\ t\\ MP_{pt}\leq\sum\limits_{m=1}^{M}\sum\limits_{\alpha_{m}^{R}=0}G_{orm}&\forall r,\ p,\ t\\ \sum\limits_{n=1}^{N}Z_{nrt}\leq MAX&\forall\ r,\ t\\ \sum\limits_{r=1}^{R}\sum\limits_{n=1}^{N}B_{nm}\cdot Z_{nrt}\leq\sum\limits_{ n=1}^{N}B_{nm}&\forall m,\ t\\ Z_{nrt}\leq\ MAR_{nr1}&\forall r,n\\ Z_{nrt}\leq\ MAR_{nr1}&\forall r,n\\ MAR_{nrt+1}\geq Z_{nrt+1}-Z_{nrt}&\forall r,\ n,\ t=1,\ldots,T-1\\ MDR_{nrt}\geq Z_{nrt}-Z_{nrt+1}&\forall r,\ n,\ t=1,\ldots,T-1\\ PBF_{pc_{1}t}\leq\sum\limits_{r=1}^{R}\sum\limits_{m=1}^{M}\sum\limits_{ \alpha_{m}^{R}=0}G_{orm}\cdot MM_{rc}&\forall p,\ c,\ c_{1},t=1,\ldots,T-1\\ PBF_{pc_{1}t}\leq\sum\limits_{r=1}^{R}\sum\limits_{m=1}^{M}\sum\limits_{ \alpha_{m}^{R}=0}G_{orm}\cdot MM_{rc_{1}}&\forall p,\ c,\ c_{1},t=1,\ldots,T-2\\ MP_{pt}\leq\sum\limits_{c=1}^{C}\sum\limits_{c_{1}=1}^{C}PBF_{pc_{1}t}\cdot MM _{rc}&\forall p,\ r,\ t=1,\ldots,T-1\\ MP_{pt}\leq\sum\limits_{c=1}^{C}\sum\limits_{c_{1}=1}^{C}PBF_{pc_{1}T-1}\cdot MM _{rc_{1}}&\forall p,\ r\\ \sum\limits_{c=1}^{C}\sum\limits_{c_{1}=1}^{C}PBF_{pc_{1}t}=1&\forall p,\ t=1, \ldots,T-1\\ \sum\limits_{c_{1}=1}^{C}PBF_{pc_{1}t}=\sum\limits_{c_{1}=1}^{C}PBF_{pc_{1}t+1}& \forall p,\ c,\ t=1,\ldots,T-2\\ \sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}(MAR_{nrt}\cdot a_{mr}\cdot B_{nm}+ MDR_{nrt}\cdot dt_{nr}\cdot B_{nm})&\forall r,\ t\\ \hskip 142.26378 pt+\sum\limits_{p=1}^{P}\sum\limits_{\alpha_{m}^{R}=0}\left(MP_ {prt}\cdot tp_{par}\cdot p_{par}\cdot p_{p}\right)\leq\xi&\forall r,\ t\\ \hskip 142.26378 ptZ_{nrt}\leq\sum\limits_{c=1}^{C}\sum\limits_{c_{1}=1}^{C}AM_{ nc_{1}t}\cdot MM_{rc}&\forall n,\ r,\ t=1,\ldots,T-1\\ Z_{nrt}\leq\sum\limits_{c=1}^{C}\sum\limits_{c_{1}=1}^{C}AM_{nc_{1}T-1}\cdot MM _{rc_{1}}&\forall n,\ r\\ \sum\limits_{c=1}^{C}\sum\limits_{c_{1}=1}^{C}AM_{nc_{1}t}=1&\forall n,\ t=1, \ldots,T-1\\ \sum\limits_{c_{1}=1}^{C}\sum\limits_{c_{1}=1}^{C}AM_{nc_{1}ct}=\sum\limits_{ c_{1}=1}^{C}AM_{nc_{1}t+1}&\forall n,\ c,\ t=1,\ldots,T-2\\ \sum\limits_{r=1}^{R}\sum\limits_{m=1}^{M}(MAR_{nrt}\cdot a_{mr}\cdot B_{nm}+ MDR_{nrt}\cdot dt_{nr}\cdot B_{nm})&\\ \hskip 142.26378 pt+\sum\limits_{c=1}^{C}\sum\limits_{c_{1}=1}^{C}AM_{nc_{1}t}- \sum\limits_{c_{1}=1}^{C}AM_{nc_{1}t}\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t \cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t \cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t \cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t \cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t \cdot t\cdot t\cdot t\cdot\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t\cdot t \cdot t\cdot t \[\sum\limits_{r=1}^{R}\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}B_{nm} \cdot(MAR_{nrt}\cdotNTA_{mrt}+MDR_{nrt}\cdot NTD_{mrt})\] \[\cdot\sum\limits_{r=1}^{R}\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N }B_{nm} \forall t \tag{27}\] \[\cdot(MAR_{nrt}\cdot\xi_{mrt}^{MA}.at_{mrt}+MDR_{nrt}\cdot\xi_{mrt}^ {MD}.dt_{mrt})\] \[PBF_{pcc1t}\;binary \forall p,c,c_{1},t\] (28) \[AMF_{ncc1t}\;binary \forall n,c,c_{1},t\] (29) \[MP_{pt}\;binary \forall p,r,t\] (30) \[Z_{mrt},MAR_{nrt},MDR_{nrt}\;binary \forall r,n,t\] (31) \[L_{rpt}\;binary \forall r,p,n,t \tag{32}\]  
Equation (3) prevents the part processing by more RMTs simultaneously. (4) ensures the availability of a module unit \(n\) of type \(m\) on RMT \(r\), if the RMT has to process part \(p\) in period \(t\). (5) sets the disassembly of module \(n\) from RMT \(r\) if not needed for the manufacturing of part batch \(p\) in period \(t\). (6) allows using module type \(m\) to manufacture part \(p\) in period \(t\) by RMT \(r\) if it is available in that period on that RMT.