## 2 Literature Review
### Solution Approaches and Model Extensions  
The authors of [12] present a solution procedure for the GLSP-based on the local search algorithm threshold accepting. The procedure starts from a configuration defined by a fixed setup pattern and calculates the corresponding lot sizes and costs through backward oriented heuristics. New candidates are created by neighborhood operations that change the old configuration and are accepted as a new configuration if they provide an improved solution. To escape local optima, new candidates are also accepted if their solution quality is not worse than a specific threshold, while the convergence of the algorithm is ensured by lowering this threshold. In [13; 17], an improved version of this procedure is used that determines lot sizes optimally for a fixed pattern through \"dual reoptimization\" instead of heuristically for solving two extended versions of the GLSP, i.e., the GLSP with setup times and the GLSP for parallel lines, respectively. While this approach proves to be efficient for the single-line GLSP, the results of the multi-line formulation are insufficient. Therefore, [PERSON] and [PERSON] (2013) adjust the approach by aggregating the original multi-line \"master-problem\" in order to decompose it into a set of isolated single-line \"sub-problems\", which are solved by the heuristic provided in [13]. A model extension of the single-stage GLSP to multiple production stages is presented in [18]. The authors of [19] adjust this model with respect to the synchronization of the machines and present several reformulations of the problem based on variable redefinitions. In addition, three heuristic approaches are presented to solve the problem. In [20], the authors present a GLSP-based model formulation called the synchronized and integrated two-level lot sizing and scheduling problem (SITLSP), which is motivated by industrial settings, mainly of soft-drink companies where the production process involves two interdependent levels with decisions concerning raw material storage and soft drink bottling. The authors of [21] solve the SITLSP by a genetic algorithm (GA) and study single-population and multi-population approaches. Additional solution approaches for the SITLSP are studied by [22]. They apply a tabu search (TS), threshold accepting (TA), genetic algorithm (GA) and hybrid genetic algorithms using a combination of GA with TS and TA, respectively, to the problem. The authors observe that for the hybrid approaches, an execution of TA and TS over the best individual of each population results in high computational times and that a strategy where these strategies are only executed when the best individual of all populations is updated performs best. The authors of [23] propose a multi-population GA with individuals structured in ternary trees to solve data provided by a soft-drink company. A simplified model formulation of the SITLSP called two-stage multi-machine lot scheduling model (P2 SMM) is proposed by [24] where each filling line can only be connected to a single tank at the same point in time. As a relaxation approach, they first solve a single-stage version of the model and afterward determine the setup variables of the P2 SMM. In addition, they propose fix-and-relax strategies and combinations of them with a relaxation approach to solve the P2 SMM. Reference [25] study different relax-and-fix strategies to solve a model formulation similar to the single-stage P2 SMM with only one production line and the production bottleneck in the bottling stage. Reference [26] propose four different alternative single-stage formulations of the P2 SMM where the first two are based on the single-stage GLSP model with sequence-dependent setup times and costs, while the other two are asymmetric traveling salesman problem (ATSP)-based formulations with different subtour elimination constraints. Numerical experiments show that solution strategies based on the ATSP formulations perform best for the P2 SMM.