## 3 A Two-Phase Iterative Solution Methodology
### The JSPT Subproblem
#### 3.2.2. JSPT Problem Formulation
  
* The following arc sets are introduced for notational brevity: \[\begin{array}{l}1.\quad{\xi_{1}}^{\prime}=\left\{\left(\left(r\right)_{p} \left(r+1\right)_{d}\right)\in\Lambda,r\in\left\{Pick_{j}^{s}\right\},r+1\in \left\{Dep_{j}^{s}\right\},j\in J\right\}\\ 2.\quad{\xi_{2a}}^{\prime}=\left\{\left(\left(r\right)_{d}\left(r\right)_{p} \right)\in\Lambda:r\in\left\{Q^{s}(j)\right\},j\in J\right\}\\ 3.\quad{\xi_{2b}}^{\prime}=\left\{\left(\left(r\right)_{d}\left(r^{\prime} \right)_{p}\right):r={\rho_{j_{1}}^{z}}^{\prime},r^{\prime}={\rho_{j_{1}}^{z}} _{|{\rho_{j}^{\prime}}^{\prime}},{\rho_{j}^{z}}\in{\rho_{j}},j\in J\right\}\\ 4.\quad{\xi_{3}}^{\prime}={\xi_{2a}}^{\prime}\cup{\xi_{2b}}^{\prime}\\ 5.\quad{\xi_{4}}^{\prime}=\left\{\left(\left(r\right)_{d}\left(r^{\prime} \right)_{p}\right)\in\Lambda:r\in\left\{Dep_{j}^{s}\right\},r^{\prime}\in \left\{Pick_{j}^{s}\right\},j,j^{\prime}\in J:j\
eq j^{\prime}\right\}\\ 6.\quad{\xi_{5}}^{\prime}=\left\{\left(O,\left(r\right)_{p}\right)\in\Lambda,r\in\left\{\left\{Q^{s}(j^{\prime})\right\}\cup{\rho_{j^{\prime}}}_{|{\rho_{j }}|}\right\},j\in J\right\}\\ \quad{\xi_{1}}^{\prime}\text{ comprises }p\text{-}d\text{ arcs within jobs that start from a {\em singleton} or the {\em last} station of a {\em group} and terminates at the succeeding {\em singleton} or the {\em first} station of the succeeding {\em group}.}\) \({\xi_{2a}}^{\prime}\text{ comprises }d\text{-}p\text{ arcs across across all {\em singletons}.}\) \({\xi_{2b}}^{\prime}\text{ comprises }\text{ordered pairs of }d\text{ and }p\text{ nodes, which denotes the {\em first} machine and {\em last} machine of the {\em group}, respectively.}\) \({\xi_{4}}^{\prime}\text{ comprises }d\text{-}p\text{ arcs across any pair of different jobs that start from a {\em singleton} or that start from the {\em first} station of a {\em group} of a particular job and terminates at the {\em singleton} or the {\em last} station of the another {\em group} of another job.}\) \({\xi_{5}}^{\prime}\text{ comprises arcs starting from }O\text{, ending at }p\text{ nodes of either a single station or the last station of a particular cluster.}\) In addition, two sets of nodes are defined for notational brevity in the following sections:  
* For each _singleton_ and the _last_ station of each machine _group_\(l\in Q^{s}(j)\cup\rho_{j_{|}\rho_{j|}}\) within each job \(j\in J\), let \(\mathrm{K}\) be the set of nodes connected to the \(p\) node of \(l\), comprising \(O\), or any of the \(d\) nodes of the dummy or processing stations from other jobs (sans its \(d\) node): \[\mathrm{K}=O\cup\left\{\cup_{j^{\prime}\in J;j^{\prime}\
eq j}\left\{Dep_{j^{ \prime}}\right\}\right\}\]2. For each _singleton_ and the _first_ station of each machine _group_\(m\in Q^{s}(j)\cup\rho_{j_{1}}\) within each job \(j\in J\), let M be the set of \(p\) nodes connected to the \(d\) node of \(l\), comprising any of the \(p\) nodes of the dummy or processing stations from other jobs (sans its \(p\) node): \[\text{M}=\left\{\cup_{j^{\prime}\in J;j^{\prime}\
eq j}\left\{\text{Pick}_{j^{ \prime}}\right\}\right\}\]  
Makespan, \(C_{max}{}^{\prime}\), is used as the objective function of \(\left|\text{SPT}\right.\) as each \(\left|\text{SPT}\right.\) is an instance of the overall FJSPT. \[MinC_{max}{}^{\prime} \tag{25}\]  
Constraint (26) enforces binary restrictions for all arcs, and Constraint (27) ensures visits to each \(p\) and \(d\) node of each \(p\)-\(d\) arc belonging to \(\varsigma_{1}{}^{\prime}\). Figure 5 illustrates an enforcement of binary restrictions. \[