## 2 Literature Review
### Related Works on MILP Model Formulation  
As mentioned in Section 1, four modeling ideas are usually adopted when formulating MILP models for shop scheduling problems: the sequence-based idea, the position-based idea, the time-based idea and the adjacent sequence-based idea [PERSON] et al. (2018). The concept behind the sequence-based modeling idea is to manage the sequential dependencies between tasks or activities. This is achieved by introducing continuous variables to represent key parameters and ensuring that tasks are completed in the correct order while meeting other time and resource limitations/constraints. Recently, [PERSON] et al. [PERSON] et al. (2018) introduced a MILP model to address the UPM scheduling problem with batch processing. For the non-permutation flow shop problem with the consideration of order acceptance and weighted tardiness, [PERSON] et al. [PERSON] et al. (2018) created a MILP model by using the sequence-based modeling idea. However, they found that medium- and large-scaled problems cannot be solved well. In addition, [PERSON] and [PERSON] and [PERSON] (2018) constructed a MILP model for minimizing the total completion time and total tardiness in a hybrid flow shop. The second modeling idea separates a workstation into several processing positions and decides which of these positions can be used for executing an operation. A MILP model for reducing the maximum completion time of a flexible job shop was initially presented by [PERSON] et al. [PERSON] et al. (2018) based on this idea. Later, [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) introduced a new MILP model by using the position-based modeling idea in a distributed scheduling system. Using the position-based modeling idea, [PERSON] et al. [PERSON] et al. (2018) constructed a MILP model for an FFSP that took various machines' energy use ratios into account. Finally, the experimental results demonstrated that there existed a trade-off relation between completion time and energy consumption. In the same way, a MILP model for an energy-efficient flexible job shop scheduling problem (FJSP) was further developed by [PERSON] et al. [PERSON] et al. (2018). The time-based modeling idea optimizes scheduling problems by dividing the total processing time into multiple periods or intervals, determining whether each operation should be executed within a specific period. This approach has been widely applied to various shop scheduling problems. For instance, [PERSON] et al. [PERSON] et al. (2018) developed a MILP for a two-phase assembly scheduling problem, introducing time-indexed decision variables to represent the start times of operations in both the process phase and the assembly phase. Similarly, [PERSON] et al. [PERSON] et al. (2018) proposed a MILP model for a proportional two-phase no-wait FJSP, noting that the solution time using CPLEX software (v.12.6.3) increased significantly with a wide range of job processing times. [PERSON] et al. [PERSON] et al. (2018) introduced a hybrid flow shop scheduling model using discrete-time representation, accounting for finite waiting times and zero intermediate capacity constraints. [PERSON] and [PERSON] [37] developed a MILP model for semiconductor final testing, extending the FJSP with finite and unstable resources. The last modeling idea involves expressing the relative sequential links between jobs in a flexible manner utilizing binary variables, time indices and restrictions. With exact order requirements, production scheduling issues can be resolved effectively. The blocking FFSP was addressed by [PERSON] et al. [38] using a MILP model, where the relationships between task families were clearly identified through decision variables. [PERSON] et al. [39] developed a MILP model that minimized the weighted total completion time and total processing cost for a UPM scheduling problem. They incorporated a binary variable to indicate whether a machine processed two distinct tasks sequentially. A novel MILP model was formulated by [PERSON] et al.