## 3 A Two-Phase Iterative Solution Methodology
### The JSPT Subproblem
#### 3.2.2. JSPT Problem Formulation
\begin{table}
\begin{tabular}{l l} \hline \hline
**Sets** & \\ \hline \(\Lambda\) & Set of arcs \\ \(\Omega\) & Set of nodes \\ \(J\) & Set of jobs \\ \(V\) & Set of transporters \\ \(N^{\prime}\) & Set of stations, \(\left|N^{\prime}\right|=\sum_{i\in J}n_{i}^{\prime}\), where \(n\eta^{\prime}\) is the number of stations of job \(j\), \(j\in J\) \\ \(N_{j}^{\prime}\) & Set of stations indexed before job \(j\), \(\left|N_{j}^{\prime}\right|=\sum\limits_{1}^{j-1}n_{j}^{\prime}\), total number of stations of the jobs indexed before job \(j\) where \\ \(n_{j}^{\prime}\) is total number of stations in a particular job \(j\). By definition, \(\left|N_{1}^{\prime}\right|=0\) \\ \(I^{\prime}\) & \(I^{\prime}=\left\{1,2,\ldots,N^{\prime}\right\}\), index set of all stations \\ \(I_{j}^{\prime}\) & \(I^{\prime}_{j}=\left\{N_{j}^{\prime\prime}+1,N_{j}^{\prime\prime}+2,\ldots N_{ j}^{\prime\prime}+n_{j}^{\prime\prime}\right\}\), set of indices in \(I^{\prime}\) associated with job \(j\) in \(J\) \\ \(O\) & Duplicated loading and unloading area \\ \(E\) & Duplicate of \(O\) \\ \(Min_{j}\) & Dummy \(p\) node for job \(j\), \(j\in J\), \(min\left(I_{j}^{\prime}\right)\) \\ \(Q(j)\) & Set of duplicate machines \\ \(Max_{j}\) & Dummy \(d\) node for job \(j\), \(j\in J\), \(max\left(I_{j}^{\prime}\right)\) \\ \(M^{\prime}(j)\) & \(M^{\prime}(j)=Min_{j}\cup Q(j)\cup Max_{j}\) \\ \(M^{\prime}\) & Set of all stations, \(M^{\prime}=\cup_{jet}\{M^{\prime}(j)\}\) \\ \((m)_{j}\) & \(p\) node(s) of any set of stations \(m\in M^{\prime}\) \\ \((m)_{d}\) & \(d\) node(s) of any set of stations \(m\in M^{\prime}\) \\ \(Pick_{j}\) & Union of \(p\) nodes for job \(j\), \(j\in J\), \(Pick_{j}=\left\{Min_{j}\cup Q(j)\right\}_{p}\) \\ \(Dep_{j}\) & Union of \(d\) nodes for job \(j\), \(j\in J\), \(Dep_{j}=\left\{Max_{j}\cup Q(j)\right\}_{d}\) \\ \(\Psi^{-}(l)\) & Set of nodes connected to \(l\in M^{\prime}\), forming arcs that directed towards \(l\), \(\Psi^{-}(l)=\left\{u:(u,l)\in\Lambda:l\
eq u\right\}\) \\ \(\Psi^{+}(l)\) & Set of nodes connected to \(l\in M^{\prime}\) forming arcs that directed away \(l\), \(\Psi^{+}(l)=\left\{v:(l,v)\in\Lambda:l\
eq v\right\}\) \\ \(\rho_{j}^{z}\) & Ordered (grouped) set of \(n\) consecutive duplicate machine \(k_{l}\), \(k_{i+1}\), \(\ldots\)\(k_{i+n-1}\) within job \(j\), \\ \(k_{i},k_{i+1},\ldots k_{i+n-1}\in Q(j)\): \(j\in J\), \(i\in I^{\prime}\), \(k_{i}<k_{i+1}<\ldots k_{i+n-1}\), \(M^{orig}(\left\{k_{i},k_{i+1},\ldots k_{i+n-1}\right\})=z,z\in M\) \\ \(\rho_{j}\) & Set of all ordered sets of \(\rho_{j}^{z}\) of job \(j\in J\), \(\rho_{j}=\cup\rho_{j}^{z}\) \\ \(\sigma_{j}\) & Set of singletons within job \(j\in J\), \(\sigma_{j}=M^{\prime}(j)\backslash\rho_{j}\) \\ \(\rho_{j_{k}}^{z}\) & \(n\)th element of \(\rho_{j}^{z}\) \\ \(\rho_{j_{k}}\) & Set of \(n\)th elements within each \(\rho_{j}^{z}\) of \(\rho_{j}\), \(\rho_{j_{k}}=\cup\rho_{j_{k}}^{z}\) \\ \(\left|\rho_{j}^{z}\right|\) & Length of respective \(\rho_{j}^{z}\) within \(\rho_{j}\). \\ \(Q^{\rm s}(j)\) & Set of singleton machines within job \(j\in J\), \(Q^{\rm s}(j)=Q(j)\backslash\rho_{j}\) \\ \(Pick_{j}^{z}\) & Union of \(p\) nodes comprising singletons and last station of each cluster within job \(j\in J\), \(Pick_{j}^{z}=\left\{Pick_{j}\backslash\rho_{j}\right\}\cup\rho_{j_{k|j}}\) \\ \(Dep_{j}^{z}\) & Union of \(d\) nodes comprising singletons and first station of each cluster within job \(j\in J\), \(Dep_{j}^{z}=\left\{Dep_{j}\backslash\rho_{j}\right\}\cup\rho_{j_{k|1}}\) \\ \(M^{dup}(k)\) & Set of machine stations duplicated from the original station \(k,k\in M\) \\ \(M^{orig}(l)\) & Original machine obtained from inverse mapping of duplicated machine, \(l\in Q(j):j\in J\) \\ \(\vartheta(k,l,m)^{z}\) & Set containing permutations of elements, \\ \(\vartheta(k,l,m)^{z}\) & \(k\in Q(j),l\in Q(j^{\prime}),m\in Q(j^{\prime}),j,j^{\prime},j^{\prime\prime} \in J:j\
eq j^{\prime}\
eq j^{\prime\prime},M^{orig}(\left\{k,l,m\right\})=z,z\in M\) \\ \(\left(\vartheta(k,l,m)^{z}\right)_{R}\) & \(\vartheta(k,l,m)^{z}:k<l,k<m\) \\ Ordered set of duplicate machines \(\cup_{l\leq i\leq n}\{k_{i}\}\) from different jobs \\ \(\chi(k_{1},k_{n})^{z}\) & \(j^{1},j^{2},\ldots j^{n}\in J:j^{1}\
eq j^{2}\
eq,\ldots j^{n},k_{i}\in Q(j^{ \prime}),1\leq i\leq n,M^{orig}(\cup_{1\leq i\leq n}\{k_{i}\})=z,z\in M\) \\ \(\lambda(k,l,m)^{z}\) & Set of duplicate machine stations \(k,l\), \(m\): \(j^{\prime}\in J:j\
eq j^{\prime},k\in Q(j),l,m\in Q(j^{\prime}),l<m\), \(M^{orig}(\left\{k,l,m\right\})=z,z\in M\) \\ \(\beta(k_{1},k_{n})^{z}\) & Ordered set of duplicate machines \(\cup_{1\leq i\leq n}\{k_{i}\}\in\cup_{j\in J}\{Q(j)\}\) where, \(k_{j}\), completes processing before the next machine, \(k_{j+1}\), starts, \(1\leq j\leq n-1\), \(M^{orig}(\cup_{1\leq i\leq n}\{k_{i}\})=z,z\in M\) \\ \hline \hline \end{tabular}
\end{table}