## 1 Introduction
### Related Research  
FLP is a well studied problem and a number of different models and algorithms have been developed. See the surveys [PERSON] and [PERSON](1985) [3], [PERSON] and [PERSON] (1987) [4], [PERSON] (1994) [5], [PERSON] and [PERSON] (1996) [6], [PERSON] et al. (2007) [7], [PERSON] et al. (2010) [8], [PERSON] and [PERSON] (2017) [9], and [PERSON] et al. (2018) [10]. Early research in this field include [PERSON] and [PERSON] (1957) [11], where the FLP is formulated as QAP (Quadratic Assignment Problem), by approximating the problem as model of assigning all facilities to different locations with the goal of minimizing the sum of the distances multiplied by the corresponding flows. In the 1990s, several exact formulation was presented by representing the candidate layout continuously (called 'continuous representation'), as opposed to the QAP formulation represents the layout candidate by a grid structure (called 'discrete representation'). According to [PERSON] and [PERSON] (2007) [12], \"By representing the FLP in a discrete fashion, the FLP is simplified, but at the penalty of eliminating many solutions from consideration. Continuous representation is more accurate and realistic than discrete representation, and thus is capable of finding the'real optimal' final layout solution.\" However, the continuous representation also increases the complexity of the FLP. In the optimization point of view, FLP belongs to the non-convex problem, and, unfortunately, there are no effective methods for solving the problem in this class. Methods for the general non-convex problem therefore take several different approaches: NLP (NonLinear Programming)-based approach; MH (Meta-Heuristics)-based approach; and MIP (Mixed-Integer-Programming)-based approach, each of which involves some _compromise_. In the NLP-based approach, FLP is represented by purely continuous variables and optimized by NLP techniques. The compromise is to give up seeking the optimal solution, and instead they seek a point that is only a locally optimal solution. As a result, the objective value of the local solution obtained is highly dependent on the choice of an initial point. Hence, much of the research is focused on a method for producing a good enough initial point. Those research include [PERSON], [PERSON] and [PERSON] (1992) [13], [PERSON] and [PERSON] (2002) [14], [PERSON] and [PERSON] (2006) [15], [PERSON], [PERSON], [PERSON] and [PERSON] (2011) [16], and [PERSON] and [PERSON] (2016) [17]. The general framework is based on the combination of two models: The first is a relaxation of the layout problem and intended to find a good initial point for the iterative algorithm used to solve the second model; the second model is an exact formulation of the facility layout problem and intended to find locally optimal solution. These methods can be fast, can handle large-scale problems, but still have the disadvantage of initial-dependency even with the relaxation step, since the relaxed model that has been proposed is _not_ convex as well. In the MH-based approach, FLP is represented by a string of finite length integer-variables and optimized by MH techniques (e.g., Simulated Annealing or Genetic Algorithm). The compromise is the quality of solution; an approximate solution is found by drawing some number of candidates from a probability distribution, and taking the best one found as the approximate solution. This approach includes LOGIC and FLEX-BAY. LOGIC is an improvement type algorithm developed by [18], where a candidate layout is represented as a slicing tree. A slicing tree is a binary tree which shows the process of recursive partitioning rectangular in such a way that each rectangular partition corresponds to the space allocated to a facility. In the LOGIC, the slicing tree is optimized using a Genetic Algorithm, and further extensions are conducted by [PERSON] and [PERSON] (1999) [19], [PERSON] and [PERSON] (2004) [20], [PERSON], [PERSON], and [PERSON] (2009) [21], and [PERSON] and [PERSON] (2017) [22].