## 3 A Two-Phase Iterative Solution Methodology
### The FJSP + T Problem  
The FJSP + T model that was first proposed by [PERSON] et al. [25] considers transportation within the FJSP by incorporating the intermachine and machine-L/U distances under the assumption of infinite transporters, resulting in no delay in transportation. Here, a simplified model is proposed through augmenting the widely cited sequence-based FJSP model of [PERSON], [PERSON] and [PERSON] [26] with transportation constraints. As [PERSON] et al. [25] introduces a dummy node to model the L/U, it possesses additional arc variables, which increases problem complexity. Unlike [PERSON] et al. [25], the proposed model focuses _solely_ on provisioning good feasible assignments for use in the subsequent phase. Machine sequencing is neglected and will often change during the solving of the associated JSPT, as it is interlinked to the transporter routing and sequencing. FJSP + T Problem Formulation The following simplifications are made to reduce solution complexity:  
1. Only intermachine transportation timings are augmented, neglecting L/U-machine distances. 2. All jobs are assumed to be _already_ located at the corresponding machines, which are used to process their first operations after assignment and are stored at the buffers of each assigned machine after job completion. 3. The notation used by FJSP + T is listed in Table 1.  
\[Min\ C_{max} \tag{1}\]  
\[C_{max}\geq C_{ik},\forall j\in J,i\in\big{|}I_{j}\big{|},k\in M_{i} \tag{2}\]  
\[C_{max}\geq\sum_{k\in M_{i}}C_{ik},\forall j\in J,i\in\big{|}I_{j}\big{|} \tag{3}\]\[S_{ik}+C_{ik}\leq U_{ik}\times F,\forall j\in J,\forall i\in I,k\in M_{i} \tag{4}\]  
\[C_{ik}\geq S_{ik}+p_{ik}-(1-U_{ik})\times F,\forall j\in J,\forall i\in I_{j},k \in M_{i} \tag{5}\]  
\[S_{ik}\geq C_{hk}-(Z_{ihk})\times F,\forall j,\forall i\
eq I,\forall i\in I_{j },\forall h\in I_{j}^{\prime},k\in M_{i}\cap M_{h} \tag{6}\]  
\[S_{hk}\geq C_{ik}-(1-Z_{ihk})\times F,\forall j\in J,\forall i\leq h,i,h\in I_{ j},k\in M_{i}\cap M_{h} \tag{7}\]  
\[\sum_{k\in M_{i}\cap M_{h}}Z_{ihk}\leq 1,\forall j,\forall j^{\prime}\in J, \forall i\in I_{j},\forall h\in I_{j}^{\prime} \tag{8}\]  
\[\sum_{l\in M_{i}}S_{il}\geq\sum_{k\in M_{i-1}}C_{i-1k},\forall j\in J,\forall i \in I_{j},2\leq i\leq|I_{j}| \tag{9}\]  
\[Min\ C_{max} \tag{1}\]0  
\[Min\ C_{max} \tag{1}\]1  
\[Min\ C_{max} \tag{1}\]2  
\[Min\ C_{max} \tag{1}\]3  
\[Min\ C_{max} \tag{1}\]4  
\[Min\ C_{max} \tag{1}\]5  
\[Min\ C_{max} \tag{1}\]6  
\[Min\ C_{max} \tag{1}\]7  
\[Min\ C_{max} \tag{1}\]8  
\[Min\ C_{max} \tag{1}\]9  
\[C_{max}\geq C_{ik},\forall j\in J,i\in\big{|}I_{j}\big{|},k\in M_{i} \tag{2}\]0  
\[C_{max}\geq C_{ik},\forall j\in J,i\in\big{|}I_{j}\big{|},k\in M_{i} \tag{2}\]1  
\[C_{max}\geq C_{ik},\forall j\in J,i\in\big{|}I_{j}\big{|},k\in M_{i} \tag{2}\]2  
\[C_{max}\geq C_{ik},\forall j\in J,i\in\big{|}I_{j}\big{|},k\in M_{i} \tag{2}\]3  
\[C_{max}\geq C_{ik},\forall j\in J,i\in\big{|}I_{j}\big{|},k\in M_{i} \tag{2}\]4Makespan (1) is primarily used as the objective function. Detailed explanations for Constraints (2)-(11) describing the FJSP model can be obtained from [PERSON], [PERSON] and [PERSON] [26]. Constraint (12) ensures the correct completion time of the assigned operation \(C_{ik}\); Constraint (13) ensures that for any two consecutive operations \(i\), \(i+1\) of a particular job \(j\), \(i,i+1\in I_{j}\) that are processed by the same machine \(k\in M_{i}\), \(i+1\) is processed immediately after i is completed. Constraint (14) computes the start time of \(i+1\) when consecutive operations \(i\), \(i+1\) of a particular job \(j\), \(i,i+1\in I_{j}\), are processed by two different machines \(k\in M_{i}\), \(l\in M_{i+1}\); Constraints (15)-(17) link \(Y_{i,i+1,k,l}\) to \(U_{ik}\) and \(U_{i+1l}\), Constraint (18) ensures that only one pair of machines \(k\in M_{i}\), \(l\in M_{i+1}\) is selected for processing. Constraint (19) is similar to conventional flow conservation constraints. Constraints (20) to (24) restrict and define the nature of the decision variables.