## 3 Related Work  
Scheduling is such a studied area that it is considered a discipline of its own. The textbook of [PERSON] [5] that is regularly updated (currently in its sixth edition) is a valuable source of information for the subject. One of the earliest contributions to machine scheduling theory and applications was [PERSON]'s influential work in [6]. The FSP is one of the most studied scheduling problems, as indicated by the many FSP reviews such as [7; 8]. This can be attributed to the fact that having jobs that should visit machines in a predefined order is a common real-world scenario in manufacturing. The PFSP, as a more constrained version of FSP that captures well the existence of conveyor belts, has also attracted much interest [9; 10]. The gap between theory and practice that was earlier identified [11] led to several extensions of the FSP and one of them is DPFSP, which is the problem investigated in this paper. Following the paper introducing the problem [1], several other papers presented new ideas and improved upon existing results [12; 13]. The recent DPFSP review paper [14] highlights that DPFSP is one of the fastest-growing topics in scheduling and that several variants of the problem have emerged, besides \(DF/prmu/C_{max}\) (i.e., classic DPFSP), such as DPFSP with extra constraints (e.g., blocking/buffer, no wait, no idle, and setup times), multi-objective DPFSP, non-deterministic DPFSP, and heterogeneous DPFSP (i.e., non-identical factories). Another recent review paper that studies distributed shop scheduling both in general and concerning DPFSP in particular can be found at [15]. Many alternatives have been proposed regarding the solution approaches to the classic DPFSP. In [1], six Mixed Integer Linear Programming (MILP) formulations were proposed, alongside two factory assignment rules and 14 heuristics based on dispatching rules. The MILP formulations were unable to operate effectively due to the resulting big sizes of  
Figure 2: The optimal solution for the first problem instance for two factories, 10 jobs, and two machines (I_2_10\(2\)1.txt), having an optimal makespan of 345 time units. the models. Still, a combination of constructive heuristics with Variable Neighborhood Descent methods resulted in good solutions. In [12], [PERSON] et al. improved the solutions by employing a scatter search algorithm. At the same time, comparisons were made with other competitive optimization methods including a Discrete Electromagnetic metaheuristic [16], a Hybrid Genetic algorithm [17], a Variable Neighborhood Descend with Branch and Bound algorithm [18], a Tabu Search algorithm [19], an Iterated Greedy algorithm [20], and others. In [13], [PERSON] et al. presented improved results, which are indeed the best known, by using Iterated Greedy algorithms employing initialization, construction, and destruction procedures, alongside local search. Comparisons were made against other approaches that also managed to produce very good results, such as a Hybrid Immune algorithm [21], a Bounded Search Iterated Greedy algorithm [22], and the Scatter Search algorithm of [12]. The recent work by [PERSON] [23] has similarities with the approach in this paper since it employs an exact algorithm in order to address the DPFSP. In particular, it uses Benders decomposition and manages to optimally solve several problems but, as the results suggest, without satisfying the same permutation of the jobs' constraint for all machines of the same factory. In [24], the total tardiness objective is examined, instead of the makespan using a mixed-integer linear programming model, heuristics, and two customized metaheuristics algorithms (i.e., a discrete Harris hawks optimization algorithm and a hybrid iterated greedy algorithm). Another relevant work is [25] that also employs DPFSP and attempts an optimization of the sum of three distinct objectives using exact methods and metaheuristics. In addition to DPFSP and its lineage, many more scheduling problems exist.