## 1 Introduction
### Related Research
FLEX-BAY is an improvement-type algorithm based on a continuous representation developed by [PERSON] and [PERSON] [23]. A layout is represented by a flexible number of vertical bays of varying width, each divided into one or more rectangular departments. Encoding flexible bay layouts is a two-part representation: permutation of the departments and breakpoints for the bays. FLEX-BAY utilizes a genetic algorithm to search the solution space by varying department-to-bay assignments or by adding or removing a bay breakpoint. A family of this approach are [PERSON], [PERSON], [PERSON] and [PERSON](2006) [24], [PERSON] and [PERSON] (2010) [25], [PERSON] and [PERSON] (2010) [26], [PERSON] and [PERSON] (2011) [27], [PERSON] et al. (2017) [28], [PERSON] et al. (2019) [29], and [PERSON] et al. (2020) [30]. Recent papers study the multi-objective model. [PERSON] et al. (2018) [31] developed a multi-objective model to minimize material handling cost, to maximize the total adjacency value, and to maximize space utilization. They applied a multi-objective particle swarm optimization. [PERSON] et al. (2019) [32] developed a multi-objective model to minimize material handling cost, to minimize the number of available workshops required for departments, and to maximize space utilization. They also applied a multi-objective particle swarm optimization. Another research trend is the dynamic and stochastic model. [PERSON] and [PERSON] (2017) [33] developed a dynamic facility layout problem based on an open queuing network theory. [PERSON] and [PERSON] (2018) [34] introduced a hybrid heuristic algorithm based on bacterial foraging optimization. [PERSON] et al. (2017) [35] developed a stochastic dynamic facility layout problem. Several studies integrated these multi-objective and dynamic models. [PERSON] et al. (2017) [36] developed a multi-objective model to minimize material handling cost, to minimize reconfiguration cost, and to minimize unsuitability between departments and location. [PERSON] and [PERSON] (2017) [37] developed a multi-objective model to minimize the total material handling cost and machine rearrangement costs, and to minimize the number of possible transporters accident that can be evaluated by the simulation. They proposed a simulation-based optimization framework in which the genetic algorithm was applied to find an optimal arrangement. While MH-based methods work very well in practice for larger sized problem instances, these approaches have a structural disadvantage in which all these heuristics cannot consider all-feasible solutions due to the encoding scheme that represents solutions as strings for the use of MH techniques. In the MIP-based approach, FLP is represented by a combination of the binary variables that specify the relative positioning of each pair of departments and the continuous variables that denote the department positioning and shapes. The outline of a general MIP-based-method is as follows. It alternates between two steps: determining a relative position of departments, and the optimizing department positions and the shapes under the specified relative positioning. If the relative positioning of the department is specified, FLP can be reduced to the convex optimization problems such as LP (Linear Programming), QP (Quadratic Programming), or SDP (SemiDefinite Programming), and, thus, can be solved by several convex optimization techniques developed in the past few decades (e.g., simplex method, active-set method, interior-point method). Unlike other two approaches which may miss the optimal solution, MIP-based approach is the true global optimization approach, and, thus, is absolutely reliable. The compromise is, however, the scalability; the first MIP-based method presented by [PERSON] in 1990 [38] can only solve the problem with six or less departments. This model is now referred to as FLP0 and several modification for MIP-FLP formulation has been conducted such as FLP1 presented by [PERSON] et al. (1999) [39]; FLP2 by [PERSON] et al.