## 3 A Two-Phase Iterative Solution Methodology
### The JSPT Subproblem
#### 3.2.2. JSPT Problem Formulation
\begin{array}{l}\omega_{kl}\leq 1,\forall(k,l)\in\varsigma_{1}\cup\varsigma _{2}\cup\varsigma_{3}\cup\varsigma_{4}\cup\varsigma_{5}:\\ \varsigma_{1}=\left\{\left(\left(r\right)_{p^{\prime}}(r+1)_{d}\right)\in \Lambda,r\in\left\{\text{Pick}_{j}\right\},j\in J\right\}\\ \varsigma_{2}=\left\{\left(\left(r\right)_{d^{\prime}}(r)_{p}\right)\in\Lambda :i_{\ell}\in Q(j),j\in J\right\}\\ \varsigma_{3}=\left\{\left(Max_{j},E\right)\in\Lambda,j\in J\right\}\\ \varsigma_{4}=\left\{\left(\left(r\right)_{d^{\prime}}(r^{\prime})_{p}\right) \in\Lambda:i\in\left\{Q(j)\right\},i^{\prime}\in\left\{\text{Pick}_{j^{\prime }}\right\},j,j^{\prime}\in J:j\
eq j^{\prime}\right\}\\ \varsigma_{5}=\left\{\left(O_{r}(r)_{p}\right)\in\Lambda,r\in\left\{\text{Pick} _{j}\right\},j\in J\right\}\\ \end{array}  \tag{26}\]  
\[\omega_{lm}=1,\forall(l,m):\left\{\left(\left(r\right)_{p^{\prime}}(r+1)_{d} \right):r\in\cup\text{Pick}_{j}^{s},j\in J\right\} \tag{27}\]  
Constraints (28) and (29) apply to \(d\) and \(p\) nodes, respectively. The constituents of each node set forming directed arcs heading into and emanating from the node of interest depend on its respective node type. \[\sum_{k\in\Psi^{-}(l)}\omega_{kl_{p}}=\omega_{l_{p}m},\forall j\in J,l\in \text{Pick}_{j},m=\left(l+1\right)_{d} \tag{28}\]  
Each \(l\) and its corresponding \(\Psi^{-}(l)\) are defined as follows:  
1. \(l=Min_{j}\Leftrightarrow\Psi^{-}(l)=\text{K}\)
2. \(l\in Q^{s}(j)\cup\rho_{j_{|\psi|}}\Leftrightarrow\Psi^{-}(l)=\text{K}\cup(l)_ {d}\)
3. \(l\in\left\{\cup\left\{\rho_{j}\backslash\rho_{j_{1}}\right\}\right\}\Leftrightarrow \Psi^{-}(l)=\left(l\right)_{d}\)  
Within (28), items (1) and (2) cater to trans-shipment and waiting; (3) specifically caters to waiting at the respective machine group. \[\omega_{kl_{d}}=\sum_{k\in\Psi^{+}(l)}\omega_{l_{d}m},\forall j\in J,m\in Dep _{j^{\prime}}k=\left(l-1\right)_{d} \tag{29}\]  
Each \(l\) and its corresponding \(\Psi^{+}(l)\) are defined as follows:  
1. \(l=Max_{j}\Leftrightarrow\Psi^{+}(l)=\text{M}\)
2. \(l\in Q^{s}(j)\cup\rho_{j_{1}}\Leftrightarrow\Psi^{+}(l)=\text{M}\cup(l)_{p}\)  
Figure 5: All possible \(p\)-\(d\) arcs combinations within each job between machine groups and machine groups (1 and 2), machine groups and singletons (2 and 3, 4 and 5), and inter-singletons (3 and 4). 3. \(l\in\left\{\cup\left\{\rho_{j}\backslash\rho_{j_{1}}\right\}\right\}\Leftrightarrow \Psi^{+}(l)=\left\{l\right\}_{p}\) Within Constraint (29), items (4) and (5) cater to trans-shipment and waiting; item (6) specifically caters to waiting at the respective machine group. Constraints (30) and (31) ensure feasibility by enforcing that for the initial visit from O, at least one dummy \(p\) node is visited, and the number of visited \(p\) nodes is capped by the available transporters. \[\sum_{l\in\left\{\cup_{jk}Min_{j}\right\}}\omega_{OI}\geq 1 \tag{30}\]  
\[\sum_{l\in\left\{\cup_{jk}Plick_{j}\right\}}\omega_{OI}\leq|V| \tag{31}\]  
Constraint (32) states that at least one transporter _returns to E_ after depositing its completed job. \[\sum_{l\in\left\{\cup_{jk}Max_{j}\right\}}\omega_{IE}\geq 1\] (32) Constraint (33) caps the number of transporters returning to E.  
\[\sum_{l\in\left\{\cup_{jk}Max_{j}\right\}}\omega_{IE}\leq|V|\] (33) Constraint (34) ensures conservation of L/U transporter. \[\sum_{l\in\left\{\cup_{jk}Max_{j}\right\}}\omega_{IE}=\sum_{l\in\left\{\cup_ {jk}Plick_{j}\right\}}\omega_{OI} \tag{34}\]  
Constraint (35) ensures that if a transporter travels from \(k\) to \(l\), then \(T_{kl}=t_{kl}\). Constraint (36) ensures that within any group, the overall processing completion time or the total waiting time of a transporter is at least equal to the total processing time of all stations.