## 3 Developed Solution Approach
### Solution Representation and Construction  
The operations-based (OB) representation scheme is commonly used in the applications of metaheuristic approaches to scheduling problems [4; 26; 31; 45]. It appears in the form of a permutation that represents the sequence by which operations will be scheduled according to an interpretation algorithm, which generally chooses the earliest start times for the operations based on the given sequence. Despite its successful implementations and reported competitive results in the literature, the OB representation is many-to-one, meaning that more than one solution representation for a specific instance can result in the same generated schedule. As indicated by [PERSON] [4], such redundancy decreases the efficiency of the search operators of a metaheuristic. Alternatively, in this paper, we propose a representation based on vectors of permutations that correspond to the workstations visiting orders for jobs and the processing sequences on machines. It is referred to as a vectors of permutations-based (VPB) representation. The proposed representation is based on the model used in [29] for the MOSP. To illustrate the proposed VPB chromosome structure, consider a simple instance of the studied problem for which the parameters are given in Tables 1-3. In this instance, there are five workstations, indexed according to \(w\in\mathcal{W}=\{1,2,3,4,5\}\). A machine is labeled as \(m_{w,i}\), where \(i\) is the assigned index of the machine in workstation \(w\). The set of all machines is denoted as \(\mathcal{M}\). Table 1 presents the structure of the system of the sample instance in terms of the machines that constitute each workstation and the ready times of the machines. There are six jobs to be scheduled, indexed according to \(j\in\mathcal{J}=\{A,B,C,D,E,F\}\). The assigned priority values for the jobs (\(p_{j}\ \forall j\in\mathcal{J}\)), their release times (\(r_{j}\ \forall j\in\mathcal{J}\)), and the sets of required workstations (\(W_{j}\subseteq\mathcal{W}\ \forall j\in\mathcal{J}\)) are listed in Table 2. Lastly, the processing times, denoted as \(d_{j,m}\), for every job \(j\in\mathcal{J}\) and machine \(m\in\mathcal{M}\), are provided in Table 3. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Workstation (\(w\))** & **Machine (\(m\))** & **Ready Time (\(a_{m}\))** \\ \hline \hline
1 & \(m_{1,1}\) & 115 \\ \(m_{1,2}\) & 0 \\ \hline
2 & \(m_{2,1}\) & 0 \\ \hline
3 & \(m_{3,1}\) & 64 \\ \hline
4 & \(m_{4,1}\) & 0 \\ \(m_{4,2}\) & 0 \\ \hline
5 & \(m_{5,1}\) & 81 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Workstations’ structure and machines’ ready times for the sample DMOSP instance. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Job (\(j\))** & **Priority (\(p_{j}\))** & **Release Time (\(r_{j}\))** & **Required Workstations (\(W_{j}\))** \\ \hline \(A\) & 8 & 28 & \{1,3,4\} \\ \(B\) & 3 & 56 & \{2,4,5\} \\ \(C\) & 5 & 43 & \{2,3,5\} \\ \(D\) & 1 & 0 & \{1,3\} \\ \(E\) & 10 & 59 & \{1,3,4,5\} \\ \(F\) & 7 & 0 & \{1,3,5\} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Jobs’ data for the sample DMOSP instance. The proposed VPB chromosome structure, denoted as \(\mathbb{C}=(\mathcal{S}_{I},\mathcal{S}_{W},\mathcal{P})\), is constituted of two strings--\(\mathcal{S}_{I}\) and \(\mathcal{S}_{W}\)--and a vector of integer values, denoted as \(\mathcal{P}\), representing locations in the string \(\mathcal{S}_{W}\). The two strings are composed of sub-strings of fixed sizes, and each sub-string is composed of elements that refer to the operations to be scheduled. Figure 2 represents the chromosome structure of a solution to the considered sample instance of the studied problem.