## 2 Methods
### Optimization Process  
This study builds upon and utilizes the same dataset as the previous research on MOO in FLP of an assembly station in a battery production factory [23]. The dataset, authorized for further study, consists of 3156 solutions generated through the MOO process. As a brief recap of the optimization process presented in [23], the initial phase involved random sampling, where it took 1506 attempts to identify 110 feasible solutions. These 110 feasible solutions formed the initial population, referred to as population zero (P(0)). After establishing P(0), the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [24] was applied, evolving the population over 10 generations (up to population ten, P(10)), to yield a set of solutions that represent various trade-offs among three conflicting objectives: space utilization, worker well-being, and distance walked by the worker (affecting production efficiency). The NSGA-II is a genetic algorithm that simulates natural evolutionary processes. It leverages biologically inspired mechanisms such as crossover and mutation to explore the solution space. Here, genes are synonymous with decision variables. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Decision Variable** & **Description** & **Datatype** \\ \hline Var. 1 & Object 1: x coordinate & Float value with 3 decimals [m] \\ Var. 2 & Object 1: y coordinate & Float value with 3 decimals [m] \\ Var. 3 & Object 2: x coordinate & Float value with 3 decimals [m] \\ Var. 4 & Object 2: y coordinate & Float value with 3 decimals [m] \\ Var. 5 & Object 3: x coordinate & Float value with 3 decimals [m] \\ Var. 6 & Object 3: y coordinate & Float value with 3 decimals [m] \\ Var. 7 & Object 3, shelf 1: z coordinate & Float value with 3 decimals [m] \\ Var. 8 & Object 3, shelf 2: z coordinate & Float value with 3 decimals [m] \\ Var. 9 & Object 4: x coordinate & Float value with 3 decimals [m] \\ Var. 10 & Object 4: y coordinate & Float value with 3 decimals [m] \\ Var. 11 & Object 4: z coordinate & Float value with 3 decimals [m] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The eleven decision variables of the rearrangeable objects for the MOO problem. Figure 1: Top view of the battery assembly station layout (**left**), showing one potential solution, with rearrangeable objects labeled as Object 1, Object 2, Object 3, and Object 4. On the (**right**), each rearrangeable object is displayed individually for clarity: Object 1 and Object 2 are pallet conveyors, Object 3 is a logistic rack, and Object 4 is a tightening tool. In NSGA-II, the optimization process begins with an initial set of solutions that is often generated randomly. This initial set of solutions, known as population zero (P(0)), constitutes a defined population size. A general rule of thumb is to have a population size approximately ten times the number of decision variables [24]. These initial solutions are also called parents. During each generation of the NSGA-II algorithm, the parent population produces new candidate solutions, called offspring, through crossover and mutation, setting up new gene/decision variable constitutions. The algorithm then combines the offspring and parent solutions to form a temporary set of solutions of double the initial population size. It applies non-dominated sorting and calculates the crowding distance to rank the temporary combined list of solutions. The top-ranked solutions that fit within the defined population size are then selected to form the next parent population for the subsequent generation. This process ensures that the best solutions survive and are carried forward to the next generation. Given the iterative nature of NSGA-II, the terms population and generation are closely related but have distinct meanings: a generation represents one iteration of the algorithm, whereas a population refers to the set of solutions at each iteration. In this extended study, the optimization process continues from population ten (P(10)).