## 2 Bi-Objective Model for the Design and Management of RMSs
### Optimization Model Design: Assumptions, Notations, and Analytic Formulation
**Parameters (Standard and OCRA)**  
\(AMAM_{mr}\) additional multiplier for the assembly of module type \(m\) on RMT \(r\)  
\(AMDM_{mr}\) additional multiplier for the disassembly of module type \(m\) on RMT \(r\)  
\(a_{pt}\) definition of the operation in which the batch of part \(p\) is in period \(t\)  
\(at_{mr}\) assembly time of module type \(m\) on RMT \(r\)\([min/module]\)  
\(B_{mn}\) 1 if auxiliary module \(n\) belongs to type \(m\); 0 otherwise \([binary]\)  
\(b_{p}\) planned production volume during a predefined period of time for part \(p\) [_parts_]  
\(DM\) duration multiplier  
\(dt_{mr}\) disassembly time of module type \(m\) from RMT \(r\)\([min/module]\)  
\(FAMAM_{mr}\) force multiplier for the assembly of module type \(m\) on RMT \(r\)  
\(FMDM_{mr}\) force multiplier for the disassembly of module type \(m\) from RMT \(r\)  
\(G_{orm}\) 1 if operation \(o\) can be performed on RMT \(r\) using an auxiliary module of type \(m\); 0 otherwise \([binary]\)  
\(CF\) constant of frequency of technical actions per minute  
\(MAX\) maximum number of modules per RMT and period [#]  
\(MM_{rc}\) 1 if RMT \(r\) is assigned to RMC \(c\); 0 otherwise \([binary]\)  
\(NTA_{mr}\) number of technical actions for the assembly of module type \(m\) on RMT \(r\)\([\#]\)  
\(NDT_{mr}\) number of technical actions for the disassembly of module type \(m\) from RMT \(r\)\([\#]\)  
\(PAMAM_{mr}\) posture multiplier for the assembly of module type \(m\) on RMT \(r\)  
\(PMDM_{mr}\) posture multiplier for the disassembly of module type \(m\) from RMT \(r\)  
\(REC\) recovery period multiplier  
\(RPMA_{mr}\) repetitiveness period multiplier for the assembly of module type \(m\) on RMT \(r\)  
\(RPMD_{mr}\) repetitiveness period multiplier for the disassembly of module type \(m\) from RMT \(r\)  
\(tp_{or}\) time to process operation \(o\) on RMT \(r\)\([min/op]\)  
\(tt_{pcc_{1}}\) travel time for batch of part \(p\) from cell \(c\) to cell \(c_{1}\)\([min/batch]\)  
\(ttam_{mc_{1}}\) travel time for auxiliary module type \(m\) from RMC \(c\) to RMC \(c_{1}\)\([min/module]\)  
\(\xi\) available time per RMT \([min/machine]\)  
\(\xi_{mr}^{MA}\)\(CF\cdot FAMAM_{mr}\cdot PAM_{mr}\cdot RPMA_{mr}\cdot AMAM_{mr}\cdot DM\)  
\(\xi_{mr}^{MD}\)\(CF\cdot FMDM_{mr}\cdot PMDM_{mr}\cdot RPMD_{mr}\cdot AMDM_{mr}\cdot DM\)  
**Decisional Variables**  
\(AMF_{nc_{1}t}\) 1 if auxiliary module \(n\) moves from RMC \(c\) to RMC \(c_{1}\) in period \(t\); 0 otherwise \([binary]\)  
\(L_{prat}\) 1 if RMT \(r\) works part \(p\) using module \(n\) in period \(t\), 0 otherwise \([binary]\)  
\(MAR_{nrt}\) 1 if module \(n\) is assembled on RMT \(r\) in period \(t\), 0 otherwise \([binary]\)  
\(MDR_{nrt}\) 1 if module \(n\) is disassembled from RMT \(r\) in period \(t\), 0 otherwise \([binary]\)  
\(MP_{prt}\) 1 if batch of part \(p\) is processed by RMT \(r\) in period \(t\); 0 otherwise \([binary]\)  
\(PBF_{pcc_{1}t}\) 1 if batch of part \(p\) moves from RMC \(c\) to RMC \(c_{1}\) in period \(t\); 0 otherwise \([binary]\)  
\(Z_{mt}\) 1 module \(n\) is on RMT \(r\) in period \(t\); 0 otherwise \([binary]\)  
**Objective functions**  
\(min\) \(\lambda\) Total part and auxiliary modules travel time and module assembly/disassembly time \([min]\)  
\(min\) \(\mu\) OCRA-based index for the auxiliary module assembly/disassembly operations