## 3 Developed Solution Approach
### Solution Representation and Construction
The condition of infeasibility is identified, and no schedule is returned if the search loop is terminated with remaining unscheduled operations. Figure 3 shows the Gantt chart of the schedule generated for the solution representation shown in Figure 2 when Algorithm 1 is applied. Note that a feasible schedule generated via Algorithm 1 is semi-active, meaning that the start time of an operation cannot be set earlier without changing the processing order within its assigned machine and/or changing its job's visiting order to its required workstations [46]. The set of semi-active schedules includes optimal solutions for many objective functions, including the ones considered in this paper. In addition, the following proposition states an important property for the relationship between the input chromosomes of Algorithm 1 and the generated schedules. **Proposition 1**. : _For any instance of the DMOSP, a schedule generated via Algorithm 1 has one and only one input VPB chromosome._  
Proof. : The proof follows from the observation that a given chromosome \((\mathcal{S}^{0}_{J},\mathcal{S}^{0}_{W},\mathcal{P}^{0})\) is interpreted to one and only one solution pair, \((\mathbb{T}^{0},\mathbb{P}^{0})\) since, as demonstrated in Figure 2, the divisions of \(\mathcal{S}^{0}_{J}\) directly correspond to the order by which jobs visit workstations, which results in one and only one \(\mathbb{T}^{0}\), and the divisions of \(\mathcal{S}^{0}_{W}\), as defined according to the set of positions \(\mathcal{P}^{0}\), directly represent the processing sequences of operations on machines, which results in one and only one \(\mathbb{P}^{0}\). Therefore, any change in \(\mathcal{S}^{0}_{J}\), \(\mathcal{S}^{0}_{W}\) or \(\mathcal{P}^{0}\) will result in a solution pair, \((\mathbb{T}^{1},\mathbb{P}^{1})\
eq(\mathbb{T}^{0},\mathbb{P}^{0})\). Let schedules \(\mathbb{S}^{0}\) and \(\mathbb{S}^{1}\) be generated via Algorithm 1 from solution pairs \((\mathbb{T}^{0},\mathbb{P}^{0})\) and \((\mathbb{T}^{1},\mathbb{P}^{1})\), respectively, where \((\mathbb{T}^{0},\mathbb{P}^{0})\
eq(\mathbb{T}^{1},\mathbb{P}^{1})\). Since Algorithm 1 is a semi-active schedule generator that schedules operations in their orders provided in a solution pair using the earliest start time for each operation, as stated in step 16, and by definition, any operation in a semi-active schedule cannot start earlier without changing the sequence to process the operations [46], \(\mathbb{S}^{0}\
eq\mathbb{S}^{1}\). Therefore, a schedule generated via Algorithm 1 cannot have more than one input VPB chromosome. The property stated in Proposition 1 represents an advantage of the proposed VPB chromosome structure compared to the OB one in terms of eliminating redundancy from the chromosome domain to the solution space. Theoretically, this can help enhance the search efficiency by avoiding search moves that change the chromosome but do not change the  
Figure 3: Gantt chart for the generated schedule based on the solution representation shown in Figure 2 for the presented sample DMOSP instance. solution. However, some guidance is needed for the search moves of the VPB representation to avoid generating chromosomes that result in infeasible solutions. To construct solutions to be used within the proposed MOSS, the construction heuristic developed in [4] is used as is. That heuristic is based on applying different rules for selecting machines and selecting jobs to be scheduled at each iteration. Such scheduling rules tend to either improve the makespan or improve the MWFT. In addition, a random selection rule is used to allow some variability in the generated initial population. The encoding of a generated solution to its corresponding VPB chromosome is straightforward since the processing orders of operations of each job and on each machine and the jobs' visiting order to their required workstations are clearly defined in every schedule.