## 3 Developed Solution Approach
### Solution Representation and Construction
Each operation is represented by a letter that refers to its job label, as given in Table 2, followed by a number that refers to its processing workstation, as provided in Table 1. As shown in Figure 2, the two strings that constitute the chromosome structure include \(\mathcal{S}_{I}\) (the upper string), which corresponds to the orders by which jobs visit the workstations, and \(\mathcal{S}_{W}\) (the lower string), which corresponds to the operations' processing sequences in the workstations. Meanwhile, the vector \(\mathcal{P}\) defines locations for separators that divide the workstations' sub-strings into divisions representing the processing sequences on the machines. For \(\mathcal{S}_{I}\), each sub-string corresponds to a job, and its size equals the number of operations of that job. To facilitate the visual identification of sub-strings, alternating grayscale colors are used in Figure 2. For instance, for the job labeled with the letter E, there are four elements in its sub-string. The order of the elements (operations) given in the job labeled with the letter C, for instance, indicates that job C will visit workstations in the order of 2, followed by 5, followed by 3. Accordingly, the upper string shown in Figure 2 can be directly interpreted into the vector of permutations expressed as \(\mathbb{T}=(\tau_{j}:j\in\mathcal{J})\) = \((\tau_{A},\tau_{B},\tau_{C},\tau_{D},\tau_{E},\tau_{F})\) = \((\langle A4\to A3\to A1\rangle\), \(\langle B2\to B4\to B5\rangle\), \(\langle\mathbb{C}2\to\mathbb{C}5\to\mathbb{C}3\rangle\), \(\langle D3\to D1\rangle\), \(\langle\mathbb{E}4\to E1\to E3\to E5\rangle,\langle F1\to F5\to F3\rangle)\), as in [29]. Having sub-strings with fixed sizes in the proposed chromosome structure entails that moves or exchanges of the elements as part of the search procedure of a metaheuristic will be conducted only within a sub-string; meanwhile, inter sub-string moves or exchanges are not allowed. In \(\mathcal{S}_{W}\), fixed-size sub-strings are defined to represent the processing sequences of operations on the machines that belong to workstations. Each sub-string corresponds to a workstation. Since a workstation may contain more than one machine, these sub-strings are divided by what is referred to as machine separators, which are defined in the vector \(\mathcal{P}\). If workstation \(w\in\mathcal{W}\) contains the set \(M_{w}\) of machines, there will be \(|M_{w}|-1\) machine separators in its sub-string. The processing sequences of the operations on the machines \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Job (\(j\))**} & \multicolumn{6}{c}{**Machine (\(m\))**} \\ \cline{2-9}  & \(m_{1,1}\) & \(m_{1,2}\) & \(m_{2,1}\) & \(m_{3,1}\) & \(m_{4,1}\) & \(m_{4,2}\) & \(m_{5,1}\) \\ \hline \(A\) & 112 & 89 & & 36 & 76 & 54 & \\ \(B\) & & & 37 & & 73 & 57 & 36 \\ \(C\) & & & 32 & 29 & & & 37 \\ \(D\) & 109 & 95 & & 30 & & & \\ \(E\) & 120 & 105 & & 37 & 67 & 51 & 44 \\ \(F\) & 118 & 101 & & 31 & & & 51 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Operations’ processing times (\(d_{j,m}\)) for the sample DMOSP instance. Figure 2: Proposed solution representation for a sample solution of the presented sample DMOSP instance. in set \(M_{w}\) are defined using the divisions of the sub-string of workstation \(w\). For instance, in Figure 2, the first division of the sub-string of workstation 4 contains the operation \(E4\), which will be processed on the first machine, \(m_{4,1}\). Meanwhile, the second division contains operation \(A4\), followed by operation \(B4\), which will both be processed on machine \(m_{4,2}\) in that order.